Disseration-Final

UNIVERSIDAD DE CÁDIZ

FACULTAD DE CIENCIAS

OPEN PROBLEMS ON TOPOLOGICAL VECTOR

SPACES WITH APPLICATIONS TO INVERSE

PROBLEMS IN BIOENGINEERING

Justin Robert Hill

Open problems on topological vector spaces with

applications to inverse problems in bioengineering

Directores: Dr. Francisco Javier García Pacheco and Dr. Clemente Cobos Sánchez

Firma Doctorando

Firma del Director Firma del Director

Cádiz, October 2016

Contents

1 Abstract v

1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1.2 Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

2 Introduction ix

2.1 Scope of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

3 Preliminary results 1

3.1 Geometric preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

3.1.1 The metric projection and suns . . . . . . . . . . . . . . . . . . . . . . . . 1

3.1.2 Convexity, balancedness, and absorbance . . . . . . . . . . . . . . . . . . 6

3.1.3 Quasi-absolute convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.4 Convexity, segments, and suns . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.5 Linear boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Topological preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Diagonals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 The finest locally convex vector topology . . . . . . . . . . . . . . . . . . 12

3.2.3 Rareness and quasi-absolute convexity . . . . . . . . . . . . . . . . . . . 13

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CONTENTS

3.2.4 Comparison of norms and barrelledness . . . . . . . . . . . . . . . . . . . 14

4 Total anti-proximinality 17

4.1 Anti-proximinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Anti-proximinality in pseudo-metric spaces . . . . . . . . . . . . . . . . . 17

4.1.2 Anti-proximinality in semi-normed spaces . . . . . . . . . . . . . . . . . 19

4.1.3 Anti-proximinal convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Total anti-proximinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Total anti-proximinality in semi-normed spaces . . . . . . . . . . . . . . 22

4.2.2 Total anti-proximinality in normed spaces . . . . . . . . . . . . . . . . . 25

4.2.3 Totally anti-proximinal convex sets . . . . . . . . . . . . . . . . . . . . . . 29

5 Ricceri’s Conjecture 33

5.1 Anti-proximinal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 The weak and the quasi anti-proximinal properties . . . . . . . . . . . . 33

5.1.2 Spaces without the weak anti-proximinal property . . . . . . . . . . . . 35

5.1.3 The anti-proximinal property . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Ricceri’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.1 Weak (positive) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2.2 Quasi (positive) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.3 Intern (positive) approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Geometric characterizations of Hilbert spaces 43

6.1 The set ΠX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.1.1 Extremal structure of ΠX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.1.2 The distance to ΠH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2 Geometric characterizations of Hilbert spaces . . . . . . . . . . . . . . . . . . . . 50

6.2.1 Using diagonals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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CONTENTS

6.2.2 Using ΠX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Applications to transcranial magnetic stimulation 53

7.1 TMS coil requirements and performance . . . . . . . . . . . . . . . . . . . . . . . 54

7.1.1 Stored magnetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.1.2 Power dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.1.3 Coil Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1.4 Induced electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1.5 Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1.6 Focality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.2.1 The current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.2.2 The magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2.3 The stored energy in the coil . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2.4 The resistive power dissipation of the coil . . . . . . . . . . . . . . . . . 58

7.2.5 The electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2.6 The temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.3 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.3.1 Minimum stored magnetic energy . . . . . . . . . . . . . . . . . . . . . . 61

7.3.2 Full field maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.3.3 Reduction of the undesired stimulation . . . . . . . . . . . . . . . . . . . 63

7.3.4 Optimised current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.3.5 Optimised temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8 TMS coil design: numerical results 67

8.1 Minimum stored magnetic energy: Coil 1 . . . . . . . . . . . . . . . . . . . . . . 68

8.2 Full field maximized: Coil 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.3 Reduction of the undesired stimulation: Coil 3 . . . . . . . . . . . . . . . . . . . 70

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CONTENTS

8.4 Optimised current: Coil 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.5 Optimised temperature: Coil 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A Mathematical foundations of the physical model 75

A.1 Supporting vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A.1.1 Generalized supporting vectors . . . . . . . . . . . . . . . . . . . . . . . . 76

A.1.2 Matrix norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.1.3 The Cholesky decomposition ψT Lψ . . . . . . . . . . . . . . . . . . . . . 83

A.2 Applications to vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.2.1 Operators with null divergence . . . . . . . . . . . . . . . . . . . . . . . . 85

A.2.2 The max scalar field associated to a vector field . . . . . . . . . . . . . . 87

B Conclusions 91

B.1 Ricceri’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

B.2 TMS coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

C Publications resulting from this work 95

C.1 Off-starting publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

C.2 Ph.D. Candidate publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

C.3 Tangential publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Bibliography 97

iv

CHAPTER

1Abstract

An abstract of this dissertation in both English and Spanish will be presented first thing.

1.1 Abstract

A totally anti-proximinal subset of a vector space is a non-empty proper subset which does not

have a nearest point whatever is the norm that the vector space is endowed with. A Hausdorff

locally convex topological vector space is said to have the (weak) anti-proximinal property

if every totally anti-proximinal (absolutely) convex subset is not rare. Ricceri’s Conjecture,

posed by Prof. Biaggio Ricceri, establishes the existence of a non-complete normed space

satisfying the anti-proximinal property. In this dissertation we approach Ricceri’s Conjecture

in the positive by proving that a Hausdorff locally convex topological vector space enjoys

the weak anti-proximinal property if and only if it is barreled. As a consequence, we show

the existence of non-complete normed spaces satisfying the weak anti-proximinal property.

v

1. ABSTRACT

We also introduce a new class of convex sets called quasi-absolutely convex and show that a

Hausdorff locally convex topological vector space satisfies the weak anti-proximinal property

if and only if every totally anti-proximinal quasi-absolutely convex subset is not rare. This

provides another partial positive solution to Ricceri’s Conjecture with many applications to

the theory of partial differential equations. We also study the intrinsic structure of totally

anti-proximinal convex subsets proving, among other things, that the absolutely convex hull

of a linearly bounded totally anti-proximinal convex set must be finitely open. As a con-

sequence of this, a new characterization of barrelledness in terms of comparison of norms

is provided. Another of our advances consists of showing that a totally anti-proximinal ab-

solutely convex subset of a vector space is linearly open. We also prove that if every totally

anti-proximinal convex subset of a vector space is linearly open then Ricceri’s Conjecture

holds true. We also demonstrate that the concept of total anti-proximinality does not make

sense in the scope of pseudo-normed spaces. Falling a bit out of Ricceri’s Conjecture, we also

study some geometric properties related to the set ΠX := (x , x∗) ∈ SX × SX ∗ : x∗ (x) = 1

obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also

compute the distance of a generic element (h, k) ∈ H ⊕2 H to ΠH for H a Hilbert space. As

an application of our mathematical results, an inverse boundary element method and effi-

cient optimisation techniques were combined to produce a versatile framework to design

truly optimal TMS coils. The presented approach can be seen as an improvement of the

work introduced by Cobos Sanchez et al. where the optimality of the resulting coil solutions

was not guaranteed. This new numerical framework has been efficiently applied to produce

TMS coils with arbitrary geometry, allowing the inclusion of new coil features in the design

process, such as optimised maximum current density or reduced temperature. Even the

structural head properties have been considered to produce more realistic TMS stimulators.

Several examples of TMS coils were designed and simulated to demonstrate the validity of

the proposed approach.

vi

1.2 Resumen

1.2 Resumen

Un subconjunto totalmente anti-proximinal de un espacio vectorial es un subconjunto propio

no vacío que no tiene un punto más cercano, cualquiera que sea la norma con la que esté

dotado el espacio vectorial. Se dice que un espacio vectorial topológico localmente convexo

y de Hausdorff tiene la propiedad anti-proximinal (débil) si cada subconjunto totalmente

anti-proximinal (absolutamente) convexo es no raro. La Conjetura de Ricceri, planteada

por el profesor Biaggio Ricceri, establece la existencia de un espacio normado no com-

pleto que satisface la propiedad anti-proximinal. En esta tesis doctoral nos acercamos a

la Conjetura de Ricceri positivamente demostrando que un espacio vectorial topológico de

Hausdorff y localmente convexo goza de la propiedad anti-proximal débil si y sólo si es

tonelado. Como consecuencia, mostramos la existencia de espacios normados no completos

que satisfacen la propiedad anti-proximal débil. También introducimos una nueva clase de

conjuntos convexos llamados cuasi absolutamente convexos y demostramos que un espacio

vectorial topológico localmente convexo de Hausdorff satisface la propiedad anti-proximal

débil si y sólo si cada subconjunto casi absolutamente convexo totalmente anti-proximinal

es no raro. Esto último proporciona otra solución positiva parcial a la Conjetura de Ricceri

con muchas aplicaciones a la teoría de ecuaciones diferenciales parciales. También estu-

diamos la estructura intrínseca de los subconjuntos convexos totalmente anti-proximinales

demostrando, entre otras cosas, que la envoltura absolutamente convexa de un conjunto

convexo totalmente anti-proximinal linealmente acotado debe ser finitamente abierta. Como

consecuencia de esto, se proporciona una nueva caracterización del concepto de tonelación

en términos de comparación de normas. Otro de nuestros avances consiste en demostrar que

un subconjunto totalmente anti-proximal absolutamente convexo de un espacio vectorial es

linealmente abierto. También probamos que si cada subconjunto convexo totalmente anti-

proximinal de un espacio vectorial es linealmente abierto entonces la Conjetura de Ricceri

es verdadera. También demostramos que el concepto de anti-proximinalidad total no tiene

vii

1. ABSTRACT

sentido en el ámbito de los espacios pseudo-normados. Saliéndonos un poco de la Con-

jetura de Ricceri, estudiamos también algunas propiedades geométricas relacionadas con

el conjunto ΠX := (x , x∗) ∈ SX × SX ∗ : x∗ (x) = 1 obteniendo dos caracterizaciones de los

espacios de Hilbert en la categoría de espacios de Banach. También calculamos la distan-

cia de un elemento genérico (h, k) ∈ H ⊕2 H a ΠH para H un espacio de Hilbert. Como

aplicación de nuestros resultados matemáticos, combinamos un método de elementos de

contorno inverso y técnicas de optimización eficientes para producir un marco versátil para

diseñar bobinas TMS verdaderamente óptimas. El enfoque presentado puede ser visto como

una mejora del trabajo introducido por Cobos Sánchez et al. donde la optimalidad de las

soluciones de las bobinas resultantes no estaba garantizada. Este nuevo marco numérico

ha sido aplicado eficientemente para producir bobinas TMS con geometría arbitraria, per-

mitiendo la inclusión de nuevas características de la bobina en el proceso de diseño, tales

como la densidad de corriente máxima optimizada o la temperatura reducida. Incluso las

propiedades estructurales de la cabeza se han considerado para producir estimuladores TMS

más realistas. Se diseñaron y simularon varios ejemplos de bobinas TMS para demostrar la

validez del enfoque propuesto.

viii

CHAPTER

2Introduction

Modern Mathematics is defined as a Former First-Order Language. It finds its origins in Pro-

positional Logic. When the symbols ∈, ⊆, ∃, and ∀ are added to the syntax of Propositional

Logic, then the Zermelo-Fraenkel Axioms, together with the Axiom of Choice or weakenings

of it, give birth to what is called Modern Mathematics. The reader may recall that the four

previous symbols are not independent as, for instance, ⊆ can be derived from ∈ as follows:

A⊆ B↔∀x (x ∈ A→ x ∈ B) .

Four major areas compose Modern Mathematics: Algebra, Topology, Analysis, and Geometry.

These four areas are integrated by Category Theory.

Definition 2.0.1. A category is a pair of classes C = (ob (C ) , hom (C )) consisting of:

• A class of objects, denoted by ob (C ). The elements of ob (C ) are sets but ob (C ) is not a

set itself, it is a class.

ix

2. INTRODUCTION

• A class of morphisms, hom (C ). The elements of hom (C ) are non-empty sets denoted by

homC (A, B) for all A, B ∈ ob (C ), that is,

hom (C ) := homC (A, B) : A, B ∈ ob (C ) .

The elements of homC (A, B) are not necessarily maps or relations from A to B. Again,

hom (C ) is not a set but a class.

• For any three objects A, B, C ∈ ob (C ), a binary operation exists

homC (A, B)× homC (B, C) → homC (A, C)

( f , g) 7→ g f ,

called composition of morphisms, verifying the following two properties:

1. Associativity: For all A, B, C , D ∈ ob (C ), all f ∈ homC (A, B), all g ∈ homC (B, C),

and all h ∈ homC (C , D), (h g) f = h (g f ).

2. Identity: For every A ∈ ob (C) there exists an element IA ∈ homC (A, A), called the

identity morphism for A, such that for all B, C ∈ ob (C ), all f ∈ homC (A, B), and

all g ∈ homC (C , A), f IA = f and IA g = g.

Among others, one observation the reader may quickly notice is that homC (A, A) endowed

with the composition becomes a monoid. What really integrates the four areas previously

mentioned into a single and unique mathematical view is the concept of functor, which

will not be defined nor treated in this manuscript. Nonetheless, general theorems about

categories can be proved that apply to all four areas of Modern Mathematics providing a

general and abstract view of the tight connections between those areas. This way we have

• global concepts relative to morphisms such as monomorphisms, epimorphisms, bi-

morphisms, isomorphisms, injections, projections, etc.;

x

2.1 Scope of this work

• global concepts relative to objects such as sub-objects, quotient objects, initial objects,

final objects, etc.;

• and global concepts relative to objects and morphisms such as products, co-products,

universal properties, etc.;


The scope of this joint work can be summarized into two main objectives:

1. Proving Ricceri’s Conjecture true or finding a counter-example.

2. Designing truly optimal TMS coils.

The birth of this work begins at a meeting between one of the advisors of the PhD candidate

with Italian Mathematician Biaggio Ricceri at the Department of Mathematics and Computer

Sciences of the University of Catania. Ricceri stated a conjecture (see (28)) on the topological

structure of certain subsets of normed spaces (see (28)). As a result of working on that

conjecture, four papers came out (18; 19; 20; 21). The last three of those four papers are

a joint work between the PhD candidate and one the advisors. Those four papers contain

original results which are not directly framed in the scope of Ricceri’s Conjecture, but are

crucial towards accomplishing our approaches to the conjecture. Therefore, we have placed

those results in a preliminary chapter in this manuscript. However, we would like the reader

to beware about the originality of the results in the Preliminary Chapter.

After maintaining conversations with some members of the Department of Electronic Engin-

eering of the College of Engineering of the University of Cadiz, the PhD candidate together

with his two advisors realized that some of the results on the Ricceri’s Conjecture could

actually be applied to optimize the norm of certain matrices that represent physical vec-

xi

2. INTRODUCTION

tor magnitudes such as the electrical field or the magnetic field. The resolution of those

optimization problems afforded the possibility of designing certain Transcranial Magnetic

Stimulation Coils.

Transcranial Magnetic Stimulation (TMS) is a non-invasive technique to stimulate the brain,

which is applied to studies of cortical effective connectivity, presurgical mapping, psychi-

atric and medical conditions, such as major depressive disorder, schizophrenia, bipolar de-

pression, post-traumatic, stress disorder and obsessive-compulsive disorder, amongst others

(31).

In TMS, strong current pulses driven through a coil are used to induce an electric field stimu-

lating neurons in the cortex. The efficiency of the stimulation is determined by coil geometry,

orientation, stimulus intensity, depth of the targeted tissue and some other factors, such as,

stimulus waveform and duration.

The TMS stimulator most commonly employed is the so called round or figure-of-eight or

butterfly coil, but since the invention of TMS numerous coil geometries have been proposed

to improved the performance and spatial characteristics of the electromagnetic stimulation

(13).

The problem in TMS coil design is to find optimal positions for the multiple windings of

coils (or equivalently the current density) so as to produce fields with the desired spatial

characteristics and properties (24) (high focality, field penetration depth, low inductance,

low heat dissipation, etc.).

Similar problems to TMS coil design can be found in engineering, in which it is also required

to determine a quasi-static spatial distribution of electric currents flowing on a conductive

surface subjected to electromagnetic constraints. Some of these problems have been suc-

cessfully solved by modelling the current under search in terms of the stream function using

xii


a boundary element method (BEM). A relevant application can be found in magnetic res-

onance imaging (MRI), where gradient coils have been efficiently designed following this

technique (27),(11).

Recently, Cobos Sanchez et al. (8) have used this numerical strategy to formulate a new

TMS coil design method, in which a stream-function based current model is incorporated

into an inverse boundary element method (IBEM). In that work, the desired current distri-

bution is eventually obtained by solving an optimization problem, where a cost function or

functional formed with a weighted linear combination of all the objectives, is minimized by

using classical techniques, such as simple partial derivation.

The computational approach in Cobos Sanchez et al. (8) has demonstrated a remarkable

flexibility for the inclusion of new coil features in the design process, such as the minimiz-

ation of the magnetic stored-energy, minimization of power dissipation or minimization of

the undesired electric field induced in non target regions of the cortex.

Although this stream function IBEM has proved to design efficient TMS stimulators, unfor-

tunately it was not known how optimal these coil solutions were. Especially since the asso-

ciated optimisation problem has a maximisation part, which has to be rigorously tackled so

as to produce the most effective stimulation of the desired cortex regions.

On the other hand, applications of TMS for diagnostic and therapeutic purposes are con-

stantly growing, being often restricted by technical limitations. The versatility of stream

function IBEM therefore opens up the possibilities of overcoming some of these restrictions

with the design of a new generation of TMS stimulators with improved performance and

novel properties, such as reduced mechanical stress, minimum coil heating, optimized max-

imum current density amongst others.

Nonetheless, most of these new performance features increase the mathematical complexity

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2. INTRODUCTION

of the TMS coil design, and prompt the need to consider a robust computational framework

to rigorously describe the problem and more efficient optimisation techniques, as classical

approaches can no longer be straightforwardly applied to handle new non-linear require-

ments.

2.2 Notation

ALL vector spaces considered in this manuscript are will be over the real or the complex field.

No other field of numbers will be dealt with alongside this dissertation. In fact, K will stand

for R or C depending on the statements involved.

card (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the cardinality of A

char (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the density character of A

int (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the topological interior of M

intA (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the topological interior of M relative to A

bd (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the topological boundary of M

bdA (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .the topological boundary of M relative to A

cl (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the topological closure of M

clA (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the topological closure of M relative to A

co (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the convex hull of M

co (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed convex hull of M

aco (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the absolutely convex hull of M

aco (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed absolutely convex hull of M

span (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the linear span of M

span (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed linear span of M

inn (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .the set of inner points of M

out (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the set of outer points of M

xiv

2.2 Notation

ext (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the set of extreme points of M

smo (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the set of smooth points of M

BX (x , r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed ball of center x and radius r in X

UX (x , r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the open ball of center x and radius r in X

SX (x , r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the sphere of center x and radius r in X

BX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed unit ball in X

UX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the open unit ball in X

SX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the unit sphere in X

JX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the duality mapping of X

NA (X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the set of norm-attaining functionals on X

X ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the topological dual of X

X ∗∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the topological bidual of X

xv

CHAPTER

3Preliminary results

3.1 Geometric preliminary results

3.1.1 The metric projection and suns

Let X be a pseudo-metric space and consider A and B to be non-empty subsets of X .

• The distance from A to B is defined by d (A, B) := inf d (A× B).

• The diameter of A is defined as d (A) := sup d (A× A) and the radius of A is defined as

r (A) := d (A)/2.

• A is said to be bounded provided that d (A)<∞.

Definition 3.1.1. Let X be a pseudo-metric space and consider x , y ∈ X . The segment of

1

3. PRELIMINARY RESULTS

extremes x and y is defined as

[x , y] := z ∈ X : d (x , y) = d (x , z) + d (z, y) .

A subset C of X is said to be convex provided that [x , y] ⊆ C for all x , y ∈ C.

Several properties verified by segments and convex sets follow:

• x , y ∈ [x , y] for all x , y ∈ X .

• [x , y] = [y, x] for all x , y ∈ X .

• [x , y] = [x , z]∪ [z, y] for all z ∈ [x , y].

As well, the non-empty intersection of any family of convex sets is also convex and thus the

convex hull of a set is defined as the intersection of all convex sets containing it. Finally,

other segments and related sets usually defined are

(x , y) := [x , y] \ x , y.

(x , y] := [x , y] \ x.

The semi-straight line or ray joining two different points x and y of extreme x is

usually defined as the set [x , y]∪ z ∈ X : y ∈ (x , z).

The straight line joining two different points x and y is usually defined as the set

z ∈ X : x ∈ (z, y) ∪ [x , y]∪ z ∈ X : y ∈ (x , z).

A set-valued function on psuedo-metric spaces that will become useful is:

Definition 3.1.2. Let X be a pseudo-metric space and consider a subset A of X . The metric

2


projection of A is defined as

PA : X → P (A)

x 7→ PA (x) := a ∈ A : d (x , A) = d (x , a) .

We refer the reader to (14, Chapter 12) for a better perspective on metric projections in inner

product spaces.

Proposition 3.1.3. Let X be a pseudo-metric space and consider a non-empty proper subset A

of X . Let x ∈ X \ A and a ∈ PA (x). Then:

1. (a, x)∩ A=∅.

2. a ∈ PA (y) for all y ∈ (a, x).

If X is a metric space, y ∈ X , a ∈ PA (y), and a 6= y, then y /∈ A (which means that condition 1

is implied by condition 2 in metric spaces).

Proof. Before showing 1 and 2 we will assume that X is a metric space, y ∈ X , a ∈ PA (y),

and a 6= y . If y ∈ A, then 0 = d (y, A) = d (y, a), which implies that y = a in virtue of the

fact that d is a metric. Now we will show 1 and 2 under the original assumption that d is a

pseudo-metric.

1. If y ∈ (a, x)∩ A, then we fall in the contradiction that

d (x , A) = d (x , a) = d (x , y) + d (y, a)> d (x , y)≥ d (x , A) .

2. If y ∈ (a, x) and there exists b ∈ A with d (y, b) < d (y, a), then we fail in the contra-

3


diction that

d (x , A) = d (x , a)

= d (x , y) + d (y, a)

> d (x , y) + d (y, b)

≥ d (x , b)

≥ d (x , A) .

An immediate use of metric projections is in the concept of a sun.

Definition 3.1.4. Let X be a pseudo-metric space. A non-empty proper subset A of X is said to

be:

• an α-sun provided that for every x ∈ X \ A there exists a ray ` starting from x such that

d (x , A) = d (x , z) + d (z, A) for all z ∈ `.

• a sun provided that for every x ∈ X \ A there exists a ∈ PA (x) such that

y ∈ X : x ∈ (a, y) ∩ A=∅

and a ∈ PA (y) for all y ∈ X with x ∈ (a, y).

• a strict sun provided that for every x ∈ X \ A and every a ∈ PA (x) we have that

y ∈ X : x ∈ (a, y) ∩ A=∅

and a ∈ PA (y) for all y ∈ X with x ∈ (a, y).

4


It is well known that every strict sun is a sun, and every sun is an α-sun. Examples of strict

suns are the Thales convex sets.

Definition 3.1.5. Let X be a metric space. A convex subset C of X is said to be Thales provided

that for all a 6= b ∈ C, all y ∈ X \ a, and all x ∈ (a, y) there exists c ∈ (a, b) verifying that

d (y, b)d (y, a)

=d (x , c)d (x , a)

.

The reason why we call these convex sets Thales is because the classical Thales Theorem is

verified.

Proposition 3.1.6. Let X be a metric space and consider a non-empty proper subset A of X . If

A is a Thales convex set, then A is a sun.

Proof. Fix arbitrary elements x ∈ X \ A and a ∈ PA (x). Now fix another arbitrary element

y ∈ X with x ∈ (a, y).

• Assume that y ∈ A. In this case we have that x ∈ (a, y) ⊂ A because A is convex, which

contradicts the fact that x ∈ X \ A.

• Let b ∈ A\ a. By hypothesis there exists c ∈ (a, b) verifying that

d (y, b) =d (y, a)d (x , a)

d (x , c) .

Now simply observe that

d (y, b) =d (y, a)d (x , a)

d (x , c)≥d (y, a)d (x , a)

d (x , A) =d (y, a)d (x , a)

d (x , a) = d (y, a) .

5


3.1.2 Convexity, balancedness, and absorbance

Given a real vector space and x , y ∈ X , the segment of extremes x and y is usually defined

as [x , y] := t x + (1− t) y : t ∈ [0,1]. Rays and straight lines are defined as follows:

The semi-straight line or ray joining two different points x and y of extreme x is

defined as the set t y + (1− t) x : t > 0.

The straight line joining two different points x and y is defined as the set

t y + (1− t) x : t ∈ R .

In the last section of the preface we will show that, in fact, the linear concept and the metric

concept of segment and convexity agree on normed spaces.

Definition 3.1.7. Let X be a real or complex vector space. A non-empty subset A of X is said to

be

• convex provided that [a, b] ⊆ A for a, b ∈ A,

• balanced provided that BK (0, 1)A⊆ A, and

• absorbing provided that for all x ∈ X there exists δx > 0 such that BK (0,δx)A⊆ A.

A set that is convex and balanced at the same time is usually called absolutely convex. The

next proposition can be found in [(16),Lemma 2.4].

Proposition 3.1.8. Let X be a real or complex vector space. If M is a convex and balanced

subset of X , then M is absorbing if and only if span (M) = X .

Proof. It is pretty obvious that every absorbing set is a generator system. Conversely, assume

that M is a generator system. Let x ∈ X \0 and consider λ1, . . . ,λn ∈K and m1, . . . , mn ∈ M

6


such that x = λ1m1+ · · ·+λnmn. Because x 6= 0, we have that |λ1|+ · · ·+ |λn|> 0, and thus

we can consider

δx :=1

|λ1|+ · · ·+ |λn|.

Now, take any α ∈ K with |α| ≤ δx . We have that αx = (αλ1)m1 + · · · + (αλn)mn and

|αλ1|+ · · ·+ |αλn| ≤ 1, therefore, since M is absolutely convex, we have that αx ∈ M .

The reader may easily check that the non-empty intersection of any family of convex or

balanced sets is convex or balanced, respectively. And the intersection of any finite family of

absorbing sets is absorbing.

Given a real or complex vector space X and a non-empty subset A of X , then:

• The balanced hull of A is defined as the intersection of all balanced subsets of X con-

taining A and denoted by bl (A). Furthermore,

bl (A) = λa : λ ∈ BK, a ∈ A .

• The convex hull of A is defined as the intersection of all convex subsets of X containing

A and denoted by co (A). Furthermore,

co (A) =

¨ n∑

i=1

t iai : t i ∈ [0, 1] , ai ∈ A,n∑

i=1

t i = 1

«

.

• The absolutely convex hull of A is defined as the intersection of all convex and balanced

subsets of X containing A and denoted by aco (A). Furthermore,

aco (A) =

¨ n∑

i=1

t iai : t i ∈K, ai ∈ A,n∑

i=1

|t i| ≤ 1

«

.

7


The convex hull of a balanced set is absolutely convex but the balanced hull of a convex set

may not be convex. In notation form, co (bl (A)) = aco (A) and bl (co (A)) ⊆ aco (A).

3.1.3 Quasi-absolute convexity

The usual concepts of balancedness and absolute convexity will be non-trivially generalized

to fit our purposes.

Definition 3.1.9. Given a real or complex vector space X and a subset A of X , we will say that

A

• almost contains 0 provided that there is a ∈ A such that a− A⊆ bl (A);

• is quasi-balanced provided that there are a ∈ A and δ ∈ UK (0,1)\0 such that a−|δ|A⊆

A;

• is quasi-absolutely convex provided that it is convex, quasi-balanced, and 0 ∈ A.

The reader may easily find examples of sets almost containing 0 but not containing 0, of

quasi-balanced sets which are not balanced, and of quasi-absolutely convex sets which are

not absolutely convex.

Remark 3.1.10. Let X be a real or complex vector space and consider a subset A of X .

1. If 0 ∈ A, then A almost contains 0.

2. If A is balanced, then it is quasi-balanced.

3. If A is quasi-balanced, then it almost contains 0.

4. If A is convex and 0 ∈ A, then [0, 1]A⊆ A and A+ A= 2A.

5. If A is absolutely convex, then A is quasi-absolutely convex.

8


3.1.4 Convexity, segments, and suns

As promised earlier, we will first prove that the vector and metric definitions for convexity

and segment are equivalent on normed spaces.

Theorem 3.1.11. Let X be a normed space, A⊂ X , and x , y ∈ X . Then:

1. [x , y] = t y + (1− t) x : t ∈ [0,1];

2. A is vector convex if and only if it is metric convex.

Proof. 1. First suppose z ∈ t y + (1− t) x : t ∈ [0,1]. Then

‖x − z‖+ ‖z − y‖ = ‖x − (t y + (1− t) x)‖+ ‖t y + (1− t) x − y‖

= ‖t x − t y‖+ ‖(1− t) x − (1− t) y‖

= t ‖x − y‖+ (1− t)‖x − y‖

= ‖x − y‖

Second, suppose z ∈ [x , y].

2. Trivial.

This fact is the linchpin that allows us to prove many theorems on normed spaces from either

a metric or vector perspective. For instance:

Theorem 3.1.12. Let X be a finite dimensional normed space. If A is a bounded convex subset

of X containing 0, then A is quasi-absolutely convex.

Proof. Let Y := span (A). In accordance to (16, Theorem 2.1) we deduce that intY (A) 6= ∅.

9


Let a ∈ A and ε,τ > 0 such that BY (a,ε) ⊆ A⊆ BY (a,τ). Finally, it suffices to notice that

a−ε

τ+ εA ⊆ a−

ε

τ+ εBY (a,τ) = a−

ε

τ+ ε(a+τBY )

=τ

τ+ εa−

ετ

τ+ εBY =

τ

τ+ ε(a− εBY )

=τ

τ+ εBY (a,ε) ⊆

τ

τ+ εA⊆ A

if we take into consideration Remark 3.1.10(1).

As well:

Theorem 3.1.13. Let X be a normed space and A a non-empty proper convex subset of X . Then

A is Euclidean and thus a sun.

Proof. Consider a 6= b ∈ A, y ∈ X \ a, and x ∈ (a, y). There exists t > 1 such that

y = t x + (1− t) a. Observe that ‖(t x + (1− t) a)− a‖= t ‖x − a‖ and hence

t =‖(t x + (1− t) a)− a‖

‖x − a‖.

We have that t−1t a+ 1

t b ∈ A since A is convex. Finally

‖(t x + (1− t) a)− b‖= t

x −

t − 1t

a+1t

b

.

3.1.5 Linear boundedness

Boundedness is a concept proper of the pre-ordered spaces that can be extended to pseudo-

metric spaces and vector spaces.

10

3.2 Topological preliminary results

Definition 3.1.14. Let X be a real vector space. A subset A of X is said to be linearly bounded

provided that A does not contain rays or straight lines

The reader may quickly notice that a set is linearly bounded if and only if every segment of it

is contained in a maximal segment. As a consequence, linearly bounded sets do not contain

non-trivial vector subspaces. The converse to this last assertion does not hold even under

the hypothesis of balancedness.

Example 3.1.15. The set

(x , y) ∈ R2 : x < 0, y ∈ (−1, 0)

∪

(x , y) ∈ R2 : x > 0, y ∈ (0, 1)

∪ (0,0)

is balanced, does not contain non-trivial vector subspaces of R2, and is not linearly bounded.

Proposition 3.1.16. Let X be a real vector space. If A is an absolutely convex subset of X , then

A is linearly bounded if and only if A contains no non-trivial vector subspaces of X .

Proof. Assume that A is absolutely convex and contains no non-trvial vector subspaces of X .

Suppose to the contrary that A is not linearly bounded and consider a 6= b ∈ A such that

a+ t (b− a) : t ∈ [0,∞) ⊆ A. It is not difficult to see that R (b− a) ⊆ A.


3.2.1 Diagonals

For a topological space X the diagonal of X × X is denoted by

DX := (x , y) ∈ X × X : x = y .

11


In case X is a topological vector space, then the anti-diagonal is defined as

D−X := (x , y) ∈ X × X : x = −y .

Lemma 3.2.1. Let X be a topological vector space.

1. For every (x , y) ∈ X × X we have

(x , y) = x + y

2,

x + y2

+ x − y

2,

y − x2

.

2. DX and D−X are topologically complemented in X × X and both isomorphic to X .

Proof.

1. Immediate.

2. It suffices to notice that the linear projection

P : X × X → DX

(x , y) → P (x , y) = x+y

2 , x+y2

is continuous and (I − P) (x , y) = x−y

2 , y−x2

for all (x , y) ∈ X × X .

3.2.2 The finest locally convex vector topology

Theorem 3.2.2 (The finest locally convex vector topology). Let X be a real or complex vector

space. There exists the finest locally convex vector topology τX on X , that is, if ν is a locally

convex vector topology on X , then ν ⊆ τX .

12


An explicit proof of the previous theorem will not be presented here. Instead, we will sketch

it through a series of definitions and remarks.

Definition 3.2.3. Let X be a real or complex vector space and consider a non-empty subset A

of X .

• We say that x ∈ X is an internal point of A when for every y ∈ X , there exists δy > 0

such that x +λy ∈ A for all λ ∈

0,δy

.

• The set of internal points of A is called the linear interior of A and is denoted by inter (A).

• A is said to be linearly open provided that A= inter (A).

The linearly open sets are precisely the open the sets of the finest locally convex vector

topology.

Remark 3.2.4. Let X be a real or complex vector space. Then

τX := A∈ P (X ) \∅ : A= inter (A) ∪ ∅ .

3.2.3 Rareness and quasi-absolute convexity

Theorem 3.2.5. Let X be a real or complex topological vector space and consider a quasi-

absolutely convex subset A of X . If the absolutely convex hull of A is not rare, then A is not rare

either.

Proof. By hypothesis we may consider a ∈ A and ε ∈ (0, 1) such that a − εA ⊆ A. We will

follow several steps:

• In the first place, we will prove that a + ε ∈ co (A∪−A) ⊆ A+ A. Indeed, let b, c ∈ A

13


and t ∈ [0,1]. Notice that

a+ ε (t b+ (1− t) (−c)) = (εt) b+ (a+ ε (1− t) (−c))

∈ A+ A

in virtue of Remark 3.1.10(1).

• In the second and last place, observe that A+ A is not rare in virtue of the previous

point, and A+ A= 2A according to Remark 3.1.10(1), therefore 2A is non-rare and so

is A.

Lemma 3.2.6. Let X be a Hausdorff locally convex topological vector space. Let A be an ab-

sorbing subset of X . If X is a Baire space, then A is not rare.

Proof. Since A is absorbing we have that X =⋃

n∈N nA. By hypothesis, there exists n ∈ N so

that nA is non-rare, and so is A.

3.2.4 Comparison of norms and barrelledness

Let X be a vector space and consider two norms |·| and ‖·‖ on X . It is well known that the

following four assertions are equivalent:

• There exists K > 0 such that |·| ≤ K ‖·‖.

• The topology induced by |·| is contained in the topology induced by ‖·‖.

• The unit ball of ‖·‖, B‖·‖, is bounded in (X , |·|).

• B|·| has non-empty interior in (X ,‖·‖).

14


In particular, any of the conditions above implies that B|·| is closed in (X ,‖·‖). We will show

now that this last assertion is equivalent to all four points above only when X is barrelled.

Let X be a vector space. Let A be a non-empty subset of X . Note that if A is absorbing, then

the Minkowski functional on A, φA, is well defined. Recall that

φA (x) := inf λ > 0 : x ∈ λA

for all x ∈ X . If, in addition, A is absolutely convex, then φA is a semi-norm on X which

verifies that

UφA⊆ A⊆ BφA

.

Finally, if, on top of everything else, A is linearly bounded, then φA is a norm on X .

Lemma 3.2.7. Let X be a topological vector space. Let A be a barrel of X and denote by |·| the

semi-norm on X given by the Minkowski functional of A. Then A= B|·|.

Proof. Observe that U|·| ⊆ A ⊆ B|·|. Let x ∈ B|·| and consider any u ∈ U|·|. It is well known

that [u, x) ⊂ U|·|, therefore 1

nu+

1− 1n

x

n∈N is a sequence in A which converges to x in

the original vector topology of X . Since A is closed in that vector topology, we deduce that

x ∈ A.

With these results in mind, we can prove the following then:

Theorem 3.2.8. Let X be a normed space with norm ‖·‖. The following conditions are equi-

valent:

1. X is barrelled.

2. If |·| is a norm on X whose unit ball is closed in the topology induced by ‖·‖, then there

exists K > 0 such that |·| ≤ K ‖·‖.

15


Proof.

(1)⇒(2) Assume that X is barrelled and let |·| be a norm on X whose unit ball, B|·|, is closed

in the topology induced by ‖·‖. Finally, notice that B|·| is a barrel of X and thus it has

non-empty interior.

(2)⇒(1) Let A be any barrel of X . Notice that we may assume without any loss of generality

that A is bounded since we can intersect it with the unit ball of X . So, let us suppose

that A is bounded. Denote by |·| the norm on X given by the Minkowski functional of

A. Since A is closed in (X ,‖·‖), by Lemma 3.2.7 we have that A= B|·| and hence B|·| is

closed in (X ‖·‖). By hypothesis, there exists K > 0 such that |·| ≤ K ‖·‖, which means

that A is a neighborhood of 0 in (X ‖·‖).

16

CHAPTER

4Total anti-proximinality

4.1 Anti-proximinality

4.1.1 Anti-proximinality in pseudo-metric spaces

Definition 4.1.1. Let E be a pseudo-metric space. A non-empty proper subset A of E is said to

be anti-proximinal provided that for all e ∈ E \ A, the distance from e to A, d (e, A), is never

attained at any a ∈ A. In other words, PA (E \ A) = ∅.

Pathological phenomena always occur when dealing with awkward pseudo-metrics as there

are such spaces free of anti-proximinal subsets.

Example 4.1.2. Let X be a set with more than one point.

1. If X is endowed with the null pseudo-metric, that is, d(x , y) = 0 for all x , y ∈ X , then

no non-empty proper subset of X is anti-proximinal. Indeed, if A is a non-empty proper

17

4. TOTAL ANTI-PROXIMINALITY

subset of X and x ∈ X \A, then it suffices to consider any a ∈ A to deduce that d (x , A) =

0= d (x , a).

2. If X is endowed with the discrete metric, that is, d(x , y) = δx y for all x , y ∈ X , then

no non-empty proper subset of X is anti-proximinal. Indeed, if A is a non-empty proper

subset of X and x ∈ X \A, then it suffices to consider any a ∈ A to deduce that d (x , A) =

1= d (x , a).

In order to assure the existence of anti-proximinal sets we need to jump to relatively good

metric spaces.

Proposition 4.1.3. Let X be a metric space. If A is a proper dense subset of X , then A is anti-

proximinal in X .

Proof. Let x ∈ X \A and assume that there exists a ∈ A such that d (x , a) = d (x , A). Observe

that d (x , A) = 0 as A is dense in X , thus d(x , a) = 0 which means that x = a ∈ A since d is a

metric. This contradicts the fact that x /∈ A.

Proposition 4.1.4. Let X be a pseudo-metric space. Let A be an anti-proximinal subset of X .

Then:

1. If Y is another pseudo-metric space and f : X → Y is a surjective k-isometry, then f (A)

is anti-proximinal in Y .

2. If X is a metric space and B is a dense subset of A, then B is also anti-proximinal in X .

Proof.

1. Let y ∈ Y \ f (A) and assume we can find a ∈ A such that d (y, f (a)) = d (y, f (A)).

Since f is surjective there exists x ∈ X such that f (x) = y . Now observe that x /∈ A

and

d (x , a) = kd (y, f (a)) = kd (y, f (A)) = d (x , A) ,

18


which means that A is not anti-proximinal in X .

2. Let x ∈ X \ B and suppose there is b ∈ B such that d (x , b) = d (x , B). We will distin-

guish between two cases:

• Assume that x ∈ A. Then d (x , b) = d (x , B) = 0, which means the contradiction

that x = b ∈ A.

• Assume that x /∈ A. Then d (x , b) = d (x , B) ≤ d (x , A) ≤ d (x , b), which means

the contradiction that A is not anti-proximinal.

Proposition 4.1.5. Let X be a pseudo-metric space. Let Aii∈I be a family of anti-proximinal

subsets of X . If⋃

i∈I Ai 6= X , then⋃

i∈I Ai is anti-proximinal.

Proof. Assume to the contrary that there are x ∈ X \⋃

i∈I Ai and c ∈⋃

i∈I Ai such that

d

x ,⋃

i∈I Ai

= d (x , c). There exists j ∈ I such that c ∈ A j . Observe now that x /∈ A j and

d (x , c) = d

x ,⋃

i∈I

Ai

≤ d

x , A j

≤ d (x , c) ,

which means that A j is not anti-proximinal.

4.1.2 Anti-proximinality in semi-normed spaces

Proposition 4.1.6. Let E be a semi-normed space. If A is an anti-proximinal subset of E, then

A+ e and λA are both anti-proximinal for every e ∈ E and every λ 6= 0.

Proof. Fix arbitrary elements e ∈ E and λ 6= 0. Observe that the maps x 7→ x+e and x 7→ λx

are an isometry and a |λ|-isometry, respectively. Therefore, in accordance to Proposition

19


4.1.4 we deduce that A+ e and λA are both anti-proximinal.

In Hilbert spaces, anti-proximinal sets verify interesting properties.

Theorem 4.1.7. Let H be a Hilbert space. If A is an anti-proximinal subset of H, then A is

fundamental, that is, span (A) = H.

Proof. Suppose to the contrary that span (A) 6= H. Take any a ∈ A and any h ∈ span (A)⊥.

Notice that h+ a /∈ A and

‖(h+ a)− a‖= d (h+ a, span (A))≤ d (h+ a, A)≤ ‖(h+ a)− a‖ ,

which means that A is not anti-proximinal.

Trivial examples of anti-proximinal subsets of semi-normed spaces are the open subsets.

4.1.3 Anti-proximinal convex sets

As expected, the convex hull of an anti-proximinal set is anti-proximinal. As of today, it

is unknown whether there exists an anti-proximinal set whose balanced hull is not anti-

proximinal.

Theorem 4.1.8. Let X be a normed space. Let A be an anti-proximinal subset of X . Then:

1. co (A) is also anti-proximinal.

2. If A almost contains 0, then bl (A) and aco (A) are both anti-proximinal.

Proof.

20


1. Let x /∈ co (A) and suppose that there exist t1, . . . , tn ∈ [0,1] and a1, . . . , an ∈ A such

that t1 6= 0, t1 + · · ·+ tn = 1, and

d (x , co (A)) =

x −n∑

i=1

t iai

.

Notice that1t1

x −n∑

n=2

t iai

/∈ A

and it is not difficult to check that

d

1t1

x −n∑

n=2

t iai

, A

=

1t1

x −n∑

n=2

t iai

− a1

,


2. First off, note that aco (A) = co (bl (A)), thus in virtue of 1 of this theorem it only

suffices to show that bl (A) is anti-proximinal. Let x /∈ bl (A) and suppose that there

exists γ ∈ BK and a ∈ A such that d (x , bl (A)) = ‖x − γa‖. We will distinguish between

two cases:

• γ = 0. In this case we have that ‖x‖ = d (x , bl (A)) ≤ ‖x −λb‖ for all b ∈ A

and all λ ∈ BK. By hypothesis we can find a0 ∈ A with a0 − A ⊆ bl (A). We will

show that y /∈ A and ‖y − a0‖ = d (y, A), where y := x + a0. Assume first that

there exists b ∈ A such that x + a0 = b. Then x = − (a0 − b) ∈ −bl (A) = bl (A),

which is not possible. Now consider any b ∈ A. By hypothesis, a0 − b ∈ bl (A),

therefore ‖y − b‖ = ‖x + (a0 − b)‖ ≥ ‖x‖ = ‖y − a0‖. This shows that A is not

anti-proximinal.

21


• γ 6= 0. Observe that1γ

x /∈ A and

d

1γ

x , A

=

1γ

x − a

,


4.2 Total anti-proximinality

As we already remarked in Example 4.1.2, if a set with more than one point is endowed

with the null pseudo-metric or the discrete metric, then no non-empty proper subset of X

is anti-proximinal. Therefore it really makes no sense to define total anti-proximinality for

pseudo-metric spaces or metric spaces.

4.2.1 Total anti-proximinality in semi-normed spaces

Let X be a semi-normed space. According to (2, Theorem 2.1) we have the following:

• The set V := x ∈ X : ‖x‖= 0 is a closed vector subspace of X .

• For every v ∈ V and every x ∈ X we have that ‖v + x‖= ‖x‖.

• If A⊆ x + V for some x ∈ X , then d (y, A) = ‖y − x‖ for all y ∈ X .

• If A⊆⋃

i∈I (x i + V ) for some family x i : i ∈ I ⊆ X , then d (y, A) = infi∈I ‖y − x i‖ for

all y ∈ X , and thus d (y, A) is always attained provided that x i : i ∈ I is compact.

Lemma 4.2.1. Let X be a semi-normed space and consider the set V = x ∈ X : ‖x‖= 0. Let

A be a non-empty proper subset of X . If there exists x ∈ X \ A such that (x − A)∩ V 6= ∅, then

22


A is not anti-proximinal for the pseudo-metric given by the semi-norm.

Proof. Simply notice that

0≤ d (x , A)≤ ‖x − a‖= 0,

where a ∈ A is so that x − a ∈ V .

Remark 4.2.2. Let X be a semi-normed space. If x ∈ X \ BX , then

d (x , BX ) = ‖x‖ − 1=

x −x‖x‖

.

Indeed,

d (x , BX )≤

x −x‖x‖

= ‖x‖ − 1,

and if y ∈ BX , then

‖x‖ − 1≤ ‖x‖ − ‖y‖ ≤ |‖x‖ − ‖y‖| ≤ ‖x − y‖ .

Lemma 4.2.3. Let X be a vector space. Let A be a totally anti-proximinal subset of X . Let ‖·‖

be a semi-norm on X . If there exists e ∈ X \ A such that d‖·‖ (e, A) > 0, then d‖·‖ (e, A) is not

attained.

Proof. Assume the existence of a ∈ A so that d‖·‖ (e, A) = ‖e− a‖. Consider any norm |·| on

X such that |e− a| ≤ ‖e− a‖. Define a new norm on X given by b·c :=max ‖·‖ , |·|. Notice

that

be− ac ≥ db·c (e, A)≥ d‖·‖ (e, A) = ‖e− a‖= be− ac,

which contradicts the fact that A is totally anti-proximinal.

Now with Lemma 4.2.3 and Lemma 4.2.1 in mind, we can prove the following:

23


Proposition 4.2.4. Let X be a vector space. Let A be a totally anti-proximinal subset of X .

Let ‖·‖ be a semi-norm on X which is not a norm and consider V = x ∈ X : ‖x‖= 0. The

following conditions are equivalent:

1. A is anti-proximinal for the pseudo-metric given by the semi-norm ‖·‖.

2. For every x ∈ X \ A we have that (x − A)∩ V =∅.

Proof.

1⇒2 Let x ∈ X \A such that (x − A)∩ V 6=∅. In virtue of Lemma 4.2.1 we deduce that A is

not anti-proximinal for the pseudo-metric given by the semi-norm ‖·‖.

2⇒1 Let x ∈ X \ A. All we need to show is that d‖·‖ (x , A) is not attained. Suppose to

the contrary then that d‖·‖ (x , A) is attained. Bearing in mind Lemma 4.2.3, we may

assume that d‖·‖ (x , A) = 0. Then there exists a ∈ A such that 0= d‖·‖ (x , A) = ‖x − a‖.

This means that x − a ∈ V and hence (x − A)∩ V 6=∅.

And with Proposition 4.2.4 nailed down, we can demonstrate then that, in fact, total anti-

proximinality can not be defined for semi-normed spaces.

Corollary 4.2.5. Let X be a vector space. No non-empty proper subset of X is anti-proximinal

for every semi-norm defined on X .

Proof. Suppose the existence of a non-empty proper subset A of X which is anti-proximinal

for every semi-norm defined on X . Since every norm is a semi-norm, in particular we have

that A is totally anti-proximinal. We will construct a semi-norm on X and find an element

x ∈ X \ A such that (x − A) ∩ V 6= ∅, which will constitute a contradiction in virtue of

24


Proposition 4.2.4. By hypothesis there exist x ∈ X \A and a ∈ A. At this stage it only suffices

to consider any semi-norm on X whose set V of null-norm vectors contains K (x − a).

4.2.2 Total anti-proximinality in normed spaces

However, in normed spaces, total anti-proximinality not only makes sense but plays a fun-

damental role in the geometry of those spaces.

Definition 4.2.6. A subset A of a vector space E is said to be totally anti-proximinal when it is

anti-proximinal for every norm on E.

Theorem 4.2.7. Let E be a vector space. Let A be a non-empty proper subset of E. If A is totally

anti-proximinal in E, then A is a generator system of E, that is, span (A) = E.

Proof. Denote P := span (A) and suppose P 6= E. Let Q be an algebraical complement for

P in E, that is, P ⊕Q = E. Let ‖·‖P and ‖·‖Q be any norms on P and Q respectively, and

consider the following norm on E given by

‖p+ q‖ :=Ç

‖p‖2P + ‖q‖2Q

where p ∈ P and q ∈ Q. It is not difficult to check that d (p+ q, P) = ‖q‖Q for all p ∈ P and

q ∈Q. Note then that if a ∈ A and q ∈Q, then a+ q /∈ A and

‖q‖Q ≥ d (a+ q, A)≥ d (a+ q, P) = ‖q‖Q ,


By bearing in mind Proposition 4.1.5 and Theorem 4.1.8 we have the following remark.

Remark 4.2.8. Let X be a vector space.

25


1. If Aii∈I is a family of totally anti-proximinal subsets of X such that⋃

i∈I Ai 6= X , then⋃

i∈I AiX is also totally anti-proximinal.

2. If A is a totally anti-proximinal subset of X , then co (A) is also totally anti-proximinal.

3. If A is a totally anti-proximinal subset of X which almost contains 0, then both bl (A) and

aco (A) are totally anti-proximinal.

Remark 4.2.9. Let X be a vector space. We will denote by L to the following fields: Q if X is

real and Q+ iQ if X is complex. It is not difficult to check that if B is a Hamel basis for X , then

Y :=

l1 b1 + · · ·+ lp bp : l j ∈ L, b j ∈ B, 1≤ j ≤ p, p ∈ N

is dense in X no matter what the vector topology X is endowed with. Notice that Y is a Q-vector

space.

Proposition 4.2.10. Let X be a vector space. Let A be a non-empty linearly open subset of X .

Then A∩ Y is totally anti-proximinal but neither linearly open nor convex, where Y is the set

considered in Remark 4.2.9.

Proof. Firstly, notice that the density of Y in X endowed with the finest locally convex vector

topology implies that A∩Y 6=∅. In fact, A∩Y is dense in A. Let x ∈ X \(A∩ Y ) and a ∈ A∩Y .

Since A is linearly open, there exists q ∈ (0,1) ∩Q such that qx + (1− q) a ∈ A. Consider

p ∈ N, λ1, . . . ,λp ∈ K, and b1, . . . , bp ∈ B such that x = λ1 b1 + · · · + λp bp, where B is a

Hamel basis for X . Now let ‖·‖ be any norm on X . Again because A is linearly open we can

find l1, . . . , lp ∈ L such that

|li −λi|<‖x − a‖p ‖bi‖

for 1≤ i ≤ p

and

q

l1 b1 + · · ·+ lp bp

+ (1− q) a ∈ A∩ Y.

26


We have the following:

x −

q

l1 b1 + · · ·+ lp bp

+ (1− q) a

≤ ‖x − (qx + (1− q) a)‖

+

(qx + (1− q) a)−

q

l1 b1 + · · ·+ lp bp

+ (1− q) a

= (1− q)‖x − a‖+ q |λ1 − l1| ‖b1‖+ · · ·+ q

λp − lp

bp

< (1− q)‖x − a‖+ q ‖x − a‖

= ‖x − a‖ .

As a consequence, d‖·‖ (x , A∩ Y ) is never attained and thus A is anti-proximinal in X endowed

with the norm ‖·‖.

The reader may notice that the main ideas of the proof of Proposition 4.2.10 can be taken

advantage of to show the following more general result, which in fact is a direct consequence

of Proposition 4.1.4(2).

Proposition 4.2.11. Let X be a vector space. If A is a totally anti-proximinal subset of X and

B is a subset of A which is dense in A for any norm on X , then B is also totally anti-proximinal

in X .

This proposition will be of much use throughout the rest of the manuscript. As well, it serves

to show counter-examples to many assertions on the properties of totally anti-proximinal

sets.

Example 4.2.12.

• The intersection of totally anti-proximinal sets is not always totally anti-proximinal.

Indeed, the sets

(−1, 1)∩Q and (−1, 1)∩ (0 ∪R \Q)

27


are both totally anti-proximinal subsets of R, however their intersection is 0 which is

not totally anti-proximinal.

• Total anti-proximinality is not hereditary to vector subspaces. Indeed, let X := R2

and consider F := R× 0 and

A :=

(x , y) ∈ R× (R \ 0) : x2 + y2 ≤ 1

∪ (0, 0) .

It is not difficult to check that A is a totally anti-proximinal non-convex subset of X .

However, A∩ F = (0,0) is not a generator system of F.

• A non-linearly open totally anti-proximinal set which has internal points. Indeed,

let X be the real line and A := (0, 1) ∪ (Q∩ (0,2)). Then A is a totally anti-proximinal

subset of X such that inter (A) 6=∅ but A is not open.

Another proposition that will come in extremely handy throughout the rest of the manuscript

comes from (18):

Proposition 4.2.13. Let X be a vector space. Let A be a non-empty subset of X .

1. If A= inter (A), then A is totally anti-proximinal.

2. Conversely, if A is totally anti-proximinal, absolutely convex and contains no half-line of

X , then A= inter (A).

Proof.

1. Let ‖·‖ be any norm on X and consider x ∈ X \ A. Take any a ∈ A and consider

the straight line passing through a. By hypothesis, there is t ∈ (0,1) such that t x +

(1− t) a ∈ A. Observe now that

‖x − a‖> (1− t)‖x − a‖= ‖x − (t x + (1− t) a)‖ ≥ d‖·‖ (x , A) .

28


2. Assume the existence of a ∈ A \ inter (A). Since span (A) = X (Theorem 4.2.7), by

Lemma 3.1.8 we have that A is absorbing in X , therefore the Minkowski functional of

A in X defines a norm on X which we will denote by ‖·‖. Simply observe now that

‖a‖= 1, since otherwise a ∈ inter (A). Indeed, observe that

U‖·‖ = inter (A) ⊂ A⊂ B‖·‖.

By hypothesis, d‖·‖ (2a, A) is never attained. However,

d‖·‖ (2a, A) = d‖·‖

2a, B‖·‖

= ‖2a‖ − 1= 1= ‖2a− a‖ ,

which is a contradiction.

4.2.3 Totally anti-proximinal convex sets

Lemma 4.2.14. Let E be a vector space. If A is totally anti-proximinal subset of A contained in

the closed unit ball of a semi-norm on X , then A is actually contained in the open unit ball of

that semi-norm.

Proof. Let ‖·‖ be any semi-norm on X whose closed unit ball B‖·‖ contains A. Suppose to the

contrary that there exists a ∈ A∩ S‖·‖. By applying Remark 4.2.2 we have that

1= ‖2a− a‖ ≥ d (2a, A)≥ d

2a, B‖·‖

= ‖2a‖ − 1= 1,

which contradicts Lemma 4.2.3.

We remind the reader that a subset of a topological vector space is said to be finitely open

29


provided that its intersection with every finite dimensional subspace is open in the Euclidean

topology. According to [(20), Theorem 3.2], a set is finitely open if and only if it is linearly

open.

Theorem 4.2.15. Let X be a vector space. Suppose that A is a totally anti-proximinal convex

subset of X . Then the absolutely convex hull of A coincides with the open unit ball of the semi-

norm that it generates and hence it is finitely open.

Proof. It is well known that in this case and since A is convex, the absolutely convex hull of

A is given by co (A∪−A). Denote by ‖·‖ the semi-norm generated by the absolutely convex

hull of A. Since

A, U‖·‖ ⊆ co (A∪−A) ⊆ B‖·‖,

by applying Lemma 4.2.14 we deduce that A⊆ U‖·‖, which automatically implies in virtue of

the triangular inequality that co (A∪−A) = U‖·‖.

Corollary 4.2.16. Let X be a vector space and A a non-empty proper subset of X . If A is totally

anti-proximinal and absolutely convex, then A is finitely open.

Hence we’ve removed the hypothesis of linear boundedness from Theorem 4.2.13(2).

Though we have met the goal of this section, we can go a little bit farther. It will not get us

the complete removal of absolute convexity from Theorem 4.2.13(2), but it will get us close.

Corollary 4.2.17. Let X be a vector space. Let A be a totally anti-proximinal subset of X . If

f : X → R is non-zero and linear, then sup f (A) is never attained.

Proof. Suppose to the contrary that a ∈ A is so that f (a) = sup f (A). Take any x ∈ X \ A

such that f (x) > f (a). Consider the semi-norm on X given by ‖·‖ := | f (·)|. Since 0 <

| f (x − a)| ≤ | f (x − b)| for all b ∈ A, we deduce that d (x , A) = ‖x − a‖, which contradicts

Lemma 4.2.3.

30


Scholium 4.2.18. Let X be a vector space. If A is a totally anti-proximinal convex subset of X

such that inter (A) 6=∅, then A= inter (A), that is, A is linearly open.

Proof. Suppose to the contrary the existence of an element a ∈ A \ inter (A). Assume X

endowed with the finest locally convex vector topology. In this situation, int (A) = inter (A) 6=

∅. According to the Hahn-Banach Separation Theorem, there exists f ∈ X ∗ \ 0 such that

f (a) ≥ sup f (int (A)) = sup f (A), which indeed implies that f (a) = sup f (A). This fact

contradicts Corollary 4.2.17.

At the very end of the next chapter it is shown that the previous scholium does not hold

true if we remove the hypothesis of convexity. In fact, we will explain why the hypothesis of

absolute convexity cannot be removed from Theorem 4.2.13(2).

31

CHAPTER

5Ricceri’s Conjecture

5.1 Anti-proximinal properties

In this section we find the original anti-proximinal property established by Ricceri (see (28))

and a couple of weakenings (see (19; 20)) that are helpful to understand how to approach

Ricceri’s Conjecture. In concrete terms, we have that the anti-proximinal property implies

the quasi anti-proximinal porperty which is equivalent to the weak anti-proximinal property.

5.1.1 The weak and the quasi anti-proximinal properties

Definition 5.1.1. A Hausdorff locally convex topological vector space is said to enjoy

• the weak anti-proximinal property if every totally anti-proximinal absolutely convex sub-

set is not rare;

33

5. RICCERI’S CONJECTURE

• the quasi anti-proximinal property if every totally anti-proximinal quasi-absolutely con-

vex subset is not rare.

Theorem 5.1.2. Let X be a Hausdorff locally convex topological vector space. The following

conditions are equivalent:

1. X satisfies the weak anti-proximinal property.

2. X satisfies the quasi anti-proximinal property.

3. X is barrelled.

Proof.

1⇒ 2 If every totally anti-proximinal quasi-absolutely convex subset of X is not rare, then X

satisfies the weak anti-proximinal property since Remark 3.1.10(2) assures that every

absolutely convex subset of X is quasi-absolutely convex.

1⇒ 2 Suppose to the contrary that X is not barrelled. By hypothesis X has a barrel M with

empty interior. Consider A := inter (M), which is an absolutely convex set since M is

so. In accordance to Theorem 4.2.13(1) we deduce that A is totally anti-proximinal.

However, int (cl (A)) = int (M) =∅.

Conversely, assume that X has the weak anti-proximinal property and consider any

totally anti-proximinal quasi-absolutely convex subset A of X . We may assume without

any loss of generality that 0 ∈ A in virtue of (18, Remark 3.1(2)). By applying (18,

Remark 3.7(3)) we have that the absolutely convex hull of A is totally anti-proximinal,

therefore it will be not rare by hypothesis. Finally, Theorem 3.2.5 allows us to deduce

that A is not rare either.

3⇒ 1 Let A be a totally anti-proximinal absolutely convex subset of E. Notice that A is a

generator system of E in virtue of Theorem 4.2.7. Next, A is absorbing in view of

34


Lemma 3.1.8, therefore its closure is a barrel of E. By hypothesis, cl (A) has non-empty

interior.

Now, with the following in mind from (26; 30),

Theorem 5.1.3. (Saxon and Wilanski)

Let X be an infinite dimensional Banach space. The following conditions are equivalent:

1. X admits an infinite dimensional separable quotient.

2. There exists a non-barrelled dense subspace Y of X .

we can prove the following:

Corollary 5.1.4. Let X be an infinite dimensional Banach space admitting an infinite dimen-

sional separable quotient. Then X satisfies the weak anti-proximinal property but admits a

proper dense subspace not enjoying it.

Proof. In accordance to Theorem 5.1.3, X has a non-barrelled dense subspace Y . Now The-

orem 5.1.2 assures that X has the weak anti-proximinal property and that Y does not.

5.1.2 Spaces without the weak anti-proximinal property

This section is devoted to explicitly construct proper dense subspaces without the weak anti-

proximinal property in infinite dimensional separable Banach spaces. First, we need the

following three results from (17):

Lemma 5.1.5 (Garcia-Pacheco, (17)). Let X be an infinite dimensional separable Banach

35


space. Let

en, e∗n

n∈N ⊂ SX × X ∗ for X be a Markushevich basis for X. The linear operator

`1→ X

(tn)n∈N→∞∑

n=1

tnen

maps ω∗-closed, bounded subsets of `1 to sequentially ω-closed subsets of X . As a consequence,

the set¨∞∑

n=1

tnen : (tn)n∈N ∈ B`1

«

is closed in X , and therefore it has empty interior in X if an only if it has empty interior in its

linear span.


space. Let

en, e∗n

n∈N ⊂ SX × X ∗ for X be a Markushevich basis for X. Then the following

statements are equivalent:

1. The basis (en)n∈N is a Schauder basis equivalent to the `1-basis.

2. The operator

`1→ X

(tn)n∈N→∞∑

n=1

tnen

is an isomorphism.

3. The set¨∞∑

n=1


«

has non-empty interior.


space. There exists a normalized Markushevich basis for X which is not a Schauder basis equi-

36


valent to the `1-basis.

Then, continuing our look at Markushevich bases, we find:

Remark 5.1.8. Let X be an infinite dimensional separable Banach space and consider a Markushev-

ich basis

en, e∗n

n∈N ⊂ SX × X ∗ for X . Every point of the absolutely convex set

¨∞∑

n=1

tnen : (tn)n∈N ∈ U`1

«

is internal in¨∞∑

n=1

tnen : (tn)n∈N ∈ `1

«

.

Indeed, it is a direct consequence of the fact that U`1is open in `1.

As well, we will need the following technical lemma:

Lemma 5.1.9. Let X be a topological space. Let Z be a subset of X . If M is a subset of Z which

is closed in X , then

intZ (M) = intcl(Z) (M) .

Proof. Let x ∈ intZ (M) and consider an open set U in X such that x ∈ U ∩ Z ⊆ M . First,

we will show that U ∩ cl (Z) ⊆ cl (U ∩ Z). Let y ∈ U ∩ cl (Z) and consider any open set

V containing y . Since y ∈ cl (Z) and U ∩ V is an open neighborhood of y we have that

V ∩ (U ∩ Z) = (U ∩ V ) ∩ Z 6= ∅. As a consequence, y ∈ cl (U ∩ Z) and hence U ∩ cl (Z) ⊆

cl (U ∩ Z). Therefore, x ∈ U ∩ Z ⊆ U ∩ cl (Z) ⊆ cl (U ∩ Z) ⊆ cl (M) = M and hence x ∈

intcl(Z) (M). Conversely, let x ∈ intcl(Z) (M) and consider an open set U in X such that x ∈

U ∩ cl (Z) ⊆ M . Notice that x ∈ Z since M ⊆ Z . Therefore, x ∈ U ∩ Z ⊆ U ∩ cl (Z) ⊆ M and

hence x ∈ intZ (M).

Now we are in the right position to state and prove the main result of this section. Since

Lemma 5.1.7 tells us that every infinite dimensional separable Banach space has a Schauder

37


basis which is not equivalent to the `1-basis, we can state the result as follows:

Theorem 5.1.10. Let X be an infinite dimensional separable Banach space and consider a

Markushevich basis

en, e∗n

n∈N ⊂ SX × X ∗ for X which is not equivalent to the `1-basis. Then

¨∞∑

n=1


«

is a dense subspace of X which does not satisfy the weak anti-proximinal property.

Proof. Consider the absolutely convex set

A :=

¨∞∑

n=1

tnen : (tn)n∈N ∈ U`1

«

.

Notice that every point of A is internal in the dense subspace

Y :=

¨∞∑

n=1


«

in virtue of Remark 5.1.8. As a consequence, A is totally anti-proximinal in Y if we bear in

mind Theorem 4.2.13(1). According to Lemma 5.1.5 the set

B :=

¨∞∑

n=1


«

is closed in X . This fact, with the collaboration of Lemma 5.1.9, brings up two consequences:

• The closure of A in X is B. Indeed, it suffices to realize that the closure of A in Y is B

and that B is closed in X .

• The interior of B in X coincides with the interior of B in Y . Indeed, it is enough to take

a look at Lemma 5.1.9.

On the other hand, in accordance with Lemma 5.1.6, the fact that

en, e∗n

n∈N is not equivalent

38

5.2 Ricceri’s Conjecture

to the `1-basis implies that B has empty interior in X and so does B in Y . In other words,

A is a totally anti-proximinal absolutely convex subset of Y which is also rare. This implies

that Y does not enjoy the weak anti-proximinal property. And it is clear that Y is a dense

subspace of X .

5.1.3 The anti-proximinal property

Definition 5.1.11 (Ricceri, (28)). A Hausdorff locally convex topological vector space is said

to have the anti-proximinal property if every totally anti-proximinal convex subset is not rare.

Theorem 5.1.12. Every non-zero finite dimensional Hausdorff topological vector space enjoys

the anti-proximinal property.

Proof. Let X be any non-zero finite dimensional Hausdorff topological vector space X and

consider A to be any totally anti-proximinal convex subset of X . We may assume without

any loss of generality that 0 ∈ A. By Theorem 4.2.7 we deduce that span (A) = X . Finally, in

view of Theorem (16, Theorem 2.1) we have that int (A) 6=∅.


The most famous conjecture ever stated by Ricceri states the following:

Conjecture 5.2.1 (Ricceri’s Conjecture, (28)). There exists a non-complete normed space en-

joying the anti-proximinal property.

5.2.1 Weak (positive) approach

Theorem 5.2.2. There exists a non-complete real normed space enjoying the weak anti-proximinal

property.

39


Proof. Let X be an infinite dimensional Banach space X and consider a non-continous lin-

ear functional f : X → K. It is well known that ker ( f ) is not closed and hence not com-

plete either. Now X is complete, therefore it is barrelled. Since ker ( f ) is of countable

co-dimension in X (see (29; 32)) we deduce that ker ( f ) is also barrelled and thus it enjoys

the weak anti-proximinal property (see Theorem 5.1.2).

5.2.2 Quasi (positive) approach

The reader may notice that the previous corollary constitutes a partial positive solution to

Ricceri’s Conjecture in the following sense:

Theorem 5.2.3. If every totally anti-proximinal convex set containing 0 is quasi-absolutely

convex, then Ricceri’s Conjecture holds true.

Proof. Indeed, if every totally anti-proximinal convex set containing 0 is quasi-absolutely

convex, then the weak anti-proximinal property and the anti-proximinal property are equi-

valent, and according to (18, Theorem 1.3) there exists a non-complete normed space en-

joying the anti-proximinal property.

5.2.3 Intern (positive) approach

In the whole of this subsection we will assume that every totally anti-proximinal convex set

has an absorbing translate.

Theorem 5.2.4. Let X be a Hausdorff locally convex topological vector space. If X is a Baire

space, then X satisfies the anti-proximinal property.

Proof. Let A be a totally anti-proximinal convex subset of X . By our assumption we deduce

the existence of a translate of A which is absorbing. In virtue of Lemma 3.2.6 we have that

40


that translate of A is non-rare, and so is A.

Theorem 5.2.4 together with several classic results (and our assumption) will give us the

key to prove Ricceri’s Conjecture true.

Example 5.2.5 (Positive Solution to Ricceri’s Conjecture). There exists a non-complete normed

space satisfying the anti-proximinal property. Indeed, it suffices to consider any non-complete

normed space which is a Baire space and apply Theorem 5.2.4. For an example of a non-complete

normed space which is Baire take a look at (3, Chapter 3) where it is observed that if E is a

separable, infinite-dimensional Banach space, then E contains a dense subspace M of countably

infinite co-dimension which is a Baire space.

41

CHAPTER

6Geometric characterizations of

Hilbert spaces

6.1 The set ΠX

In (4) the authors formally introduce the set ΠX := (x , x∗) ∈ SX × SX ∗ : x∗ (x) = 1 for X

a normed space and they use it to define a modulus of the Bishop-Phelp-Bollobás property

for functionals. However, the set ΠX appears implicitly in other indices or moduli such as

the numerical index of a Banach space, since the numerical range of a continuous linear

operator T ∈ L (X ) can be rewritten as V (T ) := x∗(T (x)) : (x , x∗) ∈ ΠX . We refer the

reader to (22) for an excellent survey paper on the numerical index of a Banach space.

It is well known that if H is a Hilbert space, then its duality mapping JH is a surjective linear

isometry, and so we can identify H with H∗ via its dual map. After this identification, ΠH

turns out to be the intersection of SH × SH with the diagonal of H ×H.

43

6. GEOMETRIC CHARACTERIZATIONS OF HILBERT SPACES

6.1.1 Extremal structure of ΠX

Given a normed space X we will define the set EX := (ext(BX )× SX ∗)∪ (SX × ext(BX ∗)).

Theorem 6.1.1. Let X be a normed space. The following conditions are equivalent:

1. ΠX ⊆ EX .

2. SX = ext (BX )∪ smo (BX ).

Proof.

1. ⇒ 2. Let x ∈ SX \ext (BX ). If x /∈ smo (BX ), then there are x∗ 6= y∗ ∈ SX ∗ such that x∗ (x) =

y∗ (y) = 1. Notice that

x , x∗+y∗

2

∈ ΠX but neither x nor x∗+y∗

2 are extreme points of

their respective balls.

2. ⇒ 1. Let (x , x∗) ∈ ΠX . Assume that x /∈ ext (BX ). By hypothesis x ∈ smo (BX ). Now if

y∗, z∗ ∈ SX ∗ and x∗ = y∗+z∗

2 , then y∗ (x) = z∗ (x) = 1 which means that y∗ = x∗ by

the smothness of x .

We recall the reader that an exposed face is the set of all vectors of norm 1 at which a given

functional of norm 1 attains its norm. An edge is a maximal segment of the unit sphere

which is an exposed face.

Corollary 6.1.2. Let X be a normed space.

1. If ΠX ⊆ EX , then every edge of BX is a maximal face of BX .

2. If X is real and 2-dimensional, then ΠX ⊆ EX .

Proof.

44

6.1 The set ΠX

1. Let [x , y] ⊂ SX be an edge of BX and consider u∗ ∈ SX ∗ such that [x , y] = (u∗)−1(1)∩

BX . Suppose to the contrary that [x , y] is not a maximal face of BX , so then it must be

contained in a maximal face C . According to the Hahn-Banach Separation Theorem,

maximal faces are exposed faces, so there exists v∗ ∈ SX ∗ such that C = (v∗)−1(1)∩BX .

Note that u∗ 6= v∗ since [x , y] ( C . Finally, x+y2 ∈ SX but x+y

2 /∈ ext (BX )∪ smo (BX ).

2. If x ∈ SX \ ext (BX ), then x belongs to the interior of a segment entirely contained in

the unit sphere. Since X is real and has dimension 2, there is only one hyperplane

supporting BX on that segment, and hence x ∈ smo (BX ).

The next example shows the existence of Banach spaces which can never be equivalently

renormed to achieve that ΠX ⊆ EX . For this we will need a bit of background.

Let ω1 denote the first uncountable ordinal. The space of all bounded real-valued functions

on [0,ω1] will be denoted by `∞ (0,ω1), which becomes a Banach space endowed with the

sup norm. The subspace of `∞ (0,ω1) composed of those functions with countable support

is denoted by m0.

Theorem 6.1.3. No equivalent norm on m0 makes Πm0⊆ Em0

.

Proof. We will divide the proof in two steps:

1. Πm0* Em0

when m0 is endowed with the sup norm. Indeed, note that in this case, m0

endowed with the sup norm isometrically contains `3∞. Now observe that Theorem

6.1.1 shows that the conditionΠX ⊆ EX is an hereditary property. Finally, it is sufficient

to realize that Π`3∞* E`3

∞in virtue of Corollary 6.1.2(1).

2. Assume that m0 is endowed with any equivalent norm. In accordance to (15, Theorem

7.12), m0 endowed with any (non-necessarily equivalent) norm has a subspace which

45


is linearly isometric to m0 endowed with the sup norm. Again, the hereditariness of

the condition ΠX ⊆ EX together with 1. concludes the proof.

6.1.2 The distance to ΠH

Our final aim is at finding the distance of a generic element (h, k) ∈ H ⊕2 H to ΠH for H

a Hilbert space. In order to accomplish this we will make use of the following couple of

lemmas. However, we will first study this issue in a more general situation.

Proposition 6.1.4. Let X be a normed space and consider ΠX in X ⊕2 X ∗. Let x ∈ SX and

y∗ ∈ SX ∗ .

1. d ((x , y∗),ΠX )≤ d

y∗, x−1(1)∩ BX ∗

.

2. If y is norm-attaining, then d ((x , y∗),ΠX )≤ d

x , (y∗)−1 (1)∩ BX

.

3. |y∗(x)− 1| ≤ 2d ((x , y∗),ΠX ).

Proof.

1. Let x∗ ∈ x−1(1)∩BX ∗ , then (x , x∗) ∈ ΠX and so d ((x , y∗),ΠX )≤ ‖(x , y∗)−(x , x∗)‖2 =

‖y∗ − x∗‖, which means that d ((x , y∗),ΠX )≤ d

y∗, x−1(1)∩ BX ∗

.

2. It follows a similar proof as in 1.

46

6.1 The set ΠX

3. Let (z, z∗) ∈ ΠX . Note that

|y∗(x)− 1| = |y∗(x)− z∗(z)|

≤ |y∗(x)− z∗(x)|+ |z∗(x)− z∗(z)|

≤ ‖y∗ − z∗‖+ ‖x − z‖

≤ 2‖(x , y∗)− (z, z∗)‖2

which implies that |y∗(x)− 1| ≤ 2d ((x , y∗),ΠX ).

Corollary 6.1.5. Let X be a normed space and consider ΠX in X ⊕2 X ∗. If x ∈ SX and y∗ ∈ SX ∗

is norm-attaining, then

|y∗(x)− 1|2

≤ d ((x , y∗),ΠX )≤min

d

y∗, x−1(1)∩ BX ∗

, d

x , (y∗)−1 (1)∩ BX

.

It is time now to take care of computing the distance of a generic element (h, k) ∈ H ⊕2 H to

ΠH .

Lemma 6.1.6. Let X be a normed space. If x ∈ X \0, then d (x , SX ) =

x − x‖x‖

= |‖x‖ − 1|.

Proof. Indeed, d (x , SX )≤

x − x‖x‖

= |‖x‖ − 1| and if y ∈ SX , then

x −x‖x‖

= |‖x‖ − 1|= |‖x‖ − ‖y‖| ≤ ‖x − y‖ . (6.1.1)

Lemma 6.1.7. Let X be a normed space and assume that X = M ⊕p N with 1 ≤ p ≤∞. Fix

arbitrary elements m ∈ M and n ∈ N.

1. d (m+ n, M) = ‖n‖.

47


2. d (m+ n, SM ) =

pp

‖n‖p + |‖m‖ − 1|p if p <∞,

max ‖n‖ , |‖m‖ − 1| if p =∞.

Proof.

1. Indeed, d (m+ n, M)≤ ‖m+ n−m‖= ‖n‖ and if m′ ∈ M then

‖n‖ ≤

m−m′

p+ ‖n‖p

1p =

m+ n−m′

p for p <∞,

‖n‖ ≤max

m−m′

,‖n‖

=

m+ n−m′

p for p =∞.

2. Indeed, we may assume that m 6= 0 and attending to Equation (6.1.1) we have that

d (m+ n, SM )≤

m+ n−m‖m‖

p=

pp

‖n‖p + |‖m‖ − 1|p if p <∞,

max ‖n‖ , |‖m‖ − 1| if p =∞,

and if m′ ∈ SM then

pp

‖n‖p + |‖m‖ − 1|p ≤ pp

‖n‖p + ‖m−m′‖p =

m+ n−m′

p for p <∞,

max ‖n‖ , |‖m‖ − 1| ≤max

‖n‖ ,

m−m′

=

m+ n−m′

p for p =∞.

The reader may notice that Lemma 6.1.7(1) still holds if M and N are simply 1-complemented

in X .

Theorem 6.1.8. Let H be a Hilbert space and consider H⊕2 H. For every h, k ∈ H we have that

d ((h, k) ,DH) =‖h−k‖p

2

d

(h, k) , SDH

=

‖h−k‖22 +

‖h+k‖p2− 1

212

d

(h, k) ,p

2SDH

=

‖h−k‖22 +

‖h+k‖p2−p

2

212

48

6.1 The set ΠX

Proof. First off, notice that H ⊕2 H = DH ⊕2 D−H in virtue of Theorem 6.2.1. By applying

Lemma 6.1.7(1) we deduce that

d ((h, k) , DH) =

h− k2

,k− h

2

2=‖h− k‖p

2.

In accordance with 2. of Lemma 6.1.7 we have that

d

(h, k) , SDH

=

‖h− k‖2

2+

‖h+ k‖p

2− 1

2

12

.

Finally,

d

(h, k) ,p

2SDH

= dp

2

1p

2(h, k)

,p

2SDH

=p

2d

hp

2,

kp

2

, SDH

=p

2

‖h− k‖2

4+

‖h+ k‖2

− 1

2

12

=

‖h− k‖2

2+

‖h+ k‖p

2−p

2

2

12

As we mentioned at the beginning of this section, ΠH =p

2SDH, so we immediately deduce

the following final corollary.

Corollary 6.1.9. Let H be a Hilbert space and consider H ⊕2 H. If h, k ∈ H, then

d ((h, k) ,ΠH) =

‖h− k‖2

2+

‖h+ k‖p

2−p

2

2

12

.

49


6.2 Geometric characterizations of Hilbert spaces

6.2.1 Using diagonals

Theorem 6.2.1. Let H be a Hilbert space and consider H ⊕2 H. Then (DH)⊥ = D−H .

Proof. Let h, k ∈ H. By the Parallelogram Law we have that

‖(h, k)‖22 = ‖h‖2 + ‖k‖2

=‖h+ k‖2

2+‖h− k‖2

2

=

h+ k2

2

+

h+ k2

2

+

h− k2

2

+

h− k2

2

=

h+ k2

,h+ k

2

2

2+

h− k2

,k− h

2

2

2.

Corollary 6.2.2. Let X be a Banach space. If DX and D−X are L2-complemented in X ⊕2 X , that

is, X ⊕2 X = DX ⊕2 D−X , then X is a Hilbert space.

Proof. Indeed, it suffices to look at the proof of Theorem 6.2.1 to realize that, under these

assumptions, X verifies the Parallelogram Law and thus it is a Hilbert space.

6.2.2 Using ΠX

If H denotes a Hilbert space, then it is clear that

ΠH = (SH × SH)∩DH =p

2SDH

provided that H ×H is endowed with the ‖ · ‖2-norm.

50

6.2 Geometric characterizations of Hilbert spaces

Theorem 6.2.3. Let X be a Banach space. If there exists a vector subspace V of X ⊕2 X ∗ such

that ΠX =p

2SV , then X is a Hilbert space and V = DX .

Proof. We will divide the proof in two steps:

1. First off, we will show that X is smooth. Suppose to the contrary that X is not, then

we can find (x , x∗), (x , y∗) ∈ ΠX such that x∗ 6= y∗. Then

(0, x∗ − y∗) = (x , x∗)− (x , y∗) ∈ ΠX −ΠX ⊆ V.

Thusp

2(0, x∗ − y∗)‖x∗ − y∗‖

∈p

2SV = ΠX ,

which is impossible.

2. According to (1, Theorem 3.2) it is sufficient to show that JX (x + y) = JX (x)+ JY (y)

for all x , y ∈ SX . So fix arbitray elements x , y ∈ SX . We may assume that x and y are

linearly independent. Note that

(x + y, JX (x) + JX (y)) = (x , JX (x)) + (y, JX (y)) ∈ ΠX +ΠX ⊆ V.

Thereforep

2(x + y, JX (x) + JX (y))

Æ

‖x + y‖2 + ‖JX (x) + JX (y)‖2∈p

2SV = ΠX .

So there exists z ∈ SX such that

p2

(x + y, JX (x) + JX (y))Æ

‖x + y‖2 + ‖JX (x) + JX (y)‖2= (z, JX (z)).

This implies that

z =x + y‖x + y‖

51


and

JX

x + y‖x + y‖

=p

2JX (x) + JX (y)

Æ

‖x + y‖2 + ‖JX (x) + JX (y)‖2

. (6.2.1)

Taking norms and solving for ‖JX (x) + JX (y)‖ we obtain that

‖JX (x) + JX (y)‖= ‖x + y‖ .

Going to back to Equation (6.2.1), we deduce that

JX (x + y) = JX (x) + JY (y) .

52

CHAPTER

7Applications to transcranial

magnetic stimulation

In this chapter an inverse boundary element method and efficient optimisation techniques

are combined to produce a versatile framework to design truly optimal TMS coils. The

presented approach can be seen as an improvement of the work introduced by Cobos Sanc-

hez et al. (8) where the optimality of the resulting coil solutions was not guaranteed. In

fact, (8) is improved and extended to produce a computational optimisation framework for

designing true optimal TMS coils of arbitrary shape. The presented technique is based on the

combination of general optimisation techniques with a stream function IBEM, which permits

the modelling of most of the TMS coil performance features as convex objectives. To illus-

trate the versatility of this computational framework, novel requirements and constraints are

prototyped here, such as minimum mechanical stress, minimum coil heating or maximum

current density.

53

7. APPLICATIONS TO TRANSCRANIAL MAGNETIC STIMULATION

This new numerical framework has been efficiently applied to produce TMS coils with ar-

bitrary geometry, allowing the inclusion of new coil features in the design process, such

as optimised maximum current density or reduced temperature. Even the structural head

properties have been considered to produce more realistic TMS stimulators.

The structure of this chapter is as follows. Firstly we review some of the most relevant

requirements and parameters that described the performance of a TMS coil. Secondly an

outline of the stream function IBEM is presented, which allows to formulate the TMS coil

design as an optimisation problem.

7.1 TMS coil requirements and performance

In the following, the properties that assess the efficiency of a TMS coil are listed.

7.1.1 Stored magnetic energy

Minimum stored magnetic energy (or equivalently minimum inductance) is an important

requirement in TMS coil design, as it enables the most rapid switching possible of the TMS

fields.

7.1.2 Power dissipation

Power requirements often limit the performance of the TMS coil. An ideal TMS stimulator

should also have a low power dissipation (or equivalently low resistance) in order to reduce

the unwanted Joule heating.

54

7.1 TMS coil requirements and performance

7.1.3 Coil Heating

Similar to the power dissipation, the conduction of high current pulse through resistive coils

leads to considerable heating that can damage the coil. Solutions such as cooling systems are

required in cases intended for prolonged high-speed stimulation, adding significant weight

and bulk.

7.1.4 Induced electric field

An ideal TMS coil should produce a strong stimulation in a prescribed region, and minimum

electric field in the rest of non target regions.

More precisely, the spatial characteristics of the TMS electromagnetic stimulation can be

described with the following parameters.

7.1.5 Penetration

Or depth, d1/2, is the radial distance from the cortical surface to the deepest point where the

electric field strength is half of its maximum value on the surface.

7.1.6 Focality

There are several definitions of focality in the literature (24); in general, more focal stimu-

lation means a smaller stimulation area with the maximum field. Here, we have employed

the focality defined through the effective surface area (13)

S1/2 =V1/2

d1/2(7.1.1)

55


where V1/2 is the volume inside the brain where the stimulus is over 50% of the maximum.

This metric takes into account that it is harder to have a focal stimulus deeper in the head.

7.2 Numerical Model

7.2.1 The current density

A model of the current under search can be achieved by using a constant boundary element

method (BEM), that allows the current distribution to be defined in terms of the nodal values

of the stream function and elements of the local geometry (see (10)).

So let us assume that the surface, S ⊆ R3, on which we want to find the optimal current,

is divided into T triangular flat elements with N nodes, which are lying at each vertex of

the element. If we consider the barycenters of the mesh triangles as RT = r1, . . . , rT , the

current density at each element can be written as

J : RT ×RN → R3

(r,ψ) 7→ J(r,ψ)≈∑N

n=1ψn n(r),

(7.2.1)

whereψ= (ψ1,ψ2, . . . ,ψN )T is the vector containing the nodal values of the stream function

and n : RT → R3 are functions related to the curl of the shape functions (10) known as

current elements. In the following, ψ ∈ RN is going to be the optimization variable.

If we denote by jnx , jn

y , jnz to the Cartesian components of n, then

J(r,ψ)≈N∑

n=1

ψn n(r) =

N∑

n=1

ψn nx (r),

N∑

n=1

ψn ny(r),

N∑

n=1

ψn nz (r)

56

7.2 Numerical Model

and the absolute current density is j(ψ) = ( j(r1,ψ), . . . , j(rT ,ψ))T where

j(r,ψ) :=

√

√

√

√

N∑

n=1

ψn nx (r)

2

+

N∑

n=1

ψn ny(r)

2

+

N∑

n=1

ψn nz (r)

2

.

7.2.2 The magnetic field

The use of this current model allows the discrete formulation of all the magnitudes involved

in the problem, for instance the magnetic field at a given point is given by

B(r,ψ)≈N∑

n=1

ψnbn(r), r ∈ R3 (7.2.2)

where bn(r) = (bnx(r), bn

y(r), bnz (r)) is the magnetic induction vector produced a unit stream

function at the nth-node (10).

By applying the current model, Eq. (7.2.1), matrix equations that transformψ to the various

coil properties and objectives can be then constructed.

The magnetic field at a series of H points, rH = r1, r2, . . . , rH

bx i(rH ,ψ) = Bx i

(rH )ψ, bx i∈ RH , Bx i

∈ RH×N , x i = x , y, z. (7.2.3)

The coefficient Bx i(h, n) = bn

x i(rh), is the x i−component of the magnetic induction produced

by the current element associated to the nth-node in the prescribed hth-point in rH .

7.2.3 The stored energy in the coil

W (ψ) =ψT Lψ, L ∈ RN×N , (7.2.4)

57


where L is the inductance matrix, which is a full symmetric matrix, and since the amount of

stored magnetic energy is always a positive

ψT Lψ> 0, ∀ψ ∈ RN , ψ 6= 0 (7.2.5)

then L is positive definite.

7.2.4 The resistive power dissipation of the coil

P(ψ) =ψT Rψ, R ∈ RN×N . (7.2.6)

where R is the resistance matrix, which is symmetric and positive-definite. Moreover, the

power dissipation can be related to the current at the surface as R∝ J Tx Jx + J T

y Jy + J Tz Jz .

7.2.5 The electric field

The electric field induced in a series of M points inside of the conducting system (11), rM =

r1, r2, . . . , rM

ex i(rM ,ψ) = Ex i

(rM )ψ, ex i∈ RM , Ex i

∈ RL×N , x i = x , y, z. (7.2.7)

7.2.6 The temperature

The temperature above ambient of the coil surface (9),

t(rRT,ψ) = ΛC (ψ), t ∈ RT , Λ ∈ RT×T , C (ψ) ∈ RT . (7.2.8)

58

7.3 Problem formulation

The matrix Λ is defined completely by the geometry of the mesh (9), and C is a vector

containing the constant value of the the Joule heating coefficient at every mesh element

C (l,ψ) =ρr

kew2J2(rl ,ψ), rl ∈ RT. (7.2.9)

where w is the thickness, ke the effective thermal conductivity and ρr the resistivity of the

conducting surface (9).


Cobos Sanchez et al. (8) employed the discretized current model presented in section 7.2

to pose the TMS coil design as an optimization, in which a cost function of ψ (that con-

tains terms to control the electric and magnetic fields induced, stored magnetic energy and

power dissipation) is minimized by using classical techniques, such as simple partial de-

rivation and subsequent matrix inversion of the consequential system of linear equations.

In the following, and for sake of comparison, this type of approach used in will be noted

as partial derivation optimisation (PDO); where it is also worth stressing that PDO cannot

handle neither linear nor quadratic requirements, such as an optimised maximum current

density (27) or optimised maximum temperature, highlighting the need of more versatile

optimization techniques.

Moreover, a key issue when designing a TMS coil is to maximise the electric field induced in

the desired cortex region. In Cobos Sanchez et al. work (8), in order to produced maximal

stimulation, the stored magnetic energy and/or power dissipation (which can be both seen

as smoothing norms of the solution), are minimized for an acceptable level of strength of

the electromagnetic fields in the target volume.

Although this scheme has proved to produce TMS coils with efficient performance, it is not

59


clear how optimal these coil solutions were, and especially whether the induced electromag-

netic fields were truly maximal in the target region.

Therefore, in order to accurately handle new coil requirements and to guarantee optimality

of the resulting solutions, we have to resort to a more mathematically rigorous approach

of the optimisation problem. In this work, it is shown that by using the suggested physical

model (Section 7.2) and after suitable mathematical derivations, the TMS coil design prob-

lem can be stated in the form of a convex optimisation. More precisely, all the most relevant

design problems can be written as

min f0(ψ)

fi(ψ)≤ bi , 1≤ i ≤ m(7.3.1)

or as

max f0(ψ)

fi(ψ)≤ bi , 1≤ i ≤ m(7.3.2)

where fi : Rn→ R are convex functions for i = 0,1, · · · , m.

Equations (7.3.1) and (7.3.2) represent a quite convenient formulation of the TMS coil

design problem, as they can be straightforwardly tackled by using one of the several op-

timisation packages available for the solution of convex problems.

In this work, two main optimization schemes have been used to solve problems in Eqs.

(7.3.1) and (7.3.2)

• Singular vector analysis.

• CVX (12) a modelling system for convex optimization problems.

Moreover, it is also worth stressing, that the solution , ψ, of problems described by Eqs.

(7.3.1) and (7.3.2) is the optimal value of the stream function at the conducting surface; the

60


final wire arrangement that approximates the continuous current distribution is produced

by contouring ψ (24).

In the following, we present a set of relevant TMS coil design example cases, which have

been chosen to demonstrate the efficiency of the optimisation framework to produce truly

optimal solutions, and to illustrate its versatility to prototype many different performance

requirements and constraints.

For sake of briefness, the coil stimulation is defined by controlling the electric field. Nonethe-

less, the corresponding formulation in terms of the induced magnetic field can be analogously

obtained by interchanging E-field and B-field.

Moreover, due to the similar nature of L and R (symmetric and positive-definite matrices),

results involving the minimum magnetic stored energy TMS coil condition, can be straight-

forwardly exported for the case of designing TMS coils with minimum power dissipation.

7.3.1 Minimum stored magnetic energy

In TMS, the brain responds maximally when the induced current is perpendicular to the

sulcus (24), it is then worth considering the design of TMS stimulator capable of inducing a

maximum electric field in a given optimal direction.

Firstly we study the problem of designing a TMS coil with minimum stored energy (induct-

ance) that maximizes one component in a given of the electric field produce in a target region

formed from a distribution of H points.

max‖Eψ‖2

minψT Lψ(7.3.3)

61


where M ∈ N with N > M , L ∈ RN×N and E ∈ RM×N , which can be Ex , Ey , Ez or the E-field

matrix in any other given direction.

Equation (7.3.3) can be transformed into a more suitable form by taking into account that

L = C T C is the Cholesky decomposition of the inductance matrix L (which is symmetric and

positive-definite) and considering a new optimisation variable given by ψ = Cψ. We have

then the following equivalences:

max‖Eψ‖2

minψT Lψ⇔

max‖

EC−1

ψ‖2

min‖ψ‖2⇔

max‖

EC−1

ψ‖2

‖ψ‖2 = 1

⇔

min‖ψ‖2

‖

EC−1

ψ‖2 =

EC−1

2 .

A solution of problem defined by Eq. (7.3.3) is simply a supporting vector of EC−1 for the

‖ · ‖2-norm, which is nothing else but a singular vector associated to the largest singular

value of EC−1, that is, any normalized eigenvector associated to the largest eigenvalue of

the symmetric matrix (EC−1)T EC−1.

7.3.2 Full field maximization

In this section, we investigate the design of a minimum stored energy TMS coil capable

of inducing an electric field with maximum magnitude in a target region formed from a

distribution of H points; this particular problem can be formulated as

max‖Exψ‖2 +

Eyψ

2 + ‖Ezψ‖2

minψT Lψ(7.3.4)

62


where M ∈ N with N > M , Ex , Ey , Ez ∈ RH×N and L ∈ RN×N .

By using again the Cholesky decomposition L = C T C of the inductance matrix L and that

ψ= Cψ, we have that Problem (7.3.4) is equivalent to

max‖

EzC−1

ψ‖22 + ‖

Ey C−1

ψ‖22 + ‖

Ex C−1

ψ‖22

‖ψ‖2 = 1(7.3.5)

Solving the above problem simply consists of finding the generalized supporting vectors of

the matrices EzC−1, Ey C−1 and Ex C−1. In other words, it suffices to find the normalized

eigenvectors associated to the largest eigenvalue of the symmetric matrix (EzC−1)T EzC−1+

(Ey C−1)T Ey C−1 + (Ex C−1)T Ex C−1. The mathematical foundations on which this last fact

relies are shown in Appendix A.1.1.

It is worth recalling that the magnetic induction (and the electric field for for regions with

uniform conductivity and no electric charge) is divergence-free field , so here we would like

to make the reader notice that the problem of designing a TMS coil that maximises Bx , By

and Bz fields is not equivalent to one that maximises two components. A full proof of this

statement can be found in Appendix A.2.1.

7.3.3 Reduction of the undesired stimulation

Precise spatial localization of stimulation sites is one of the keys of an efficient TMS pro-

cedure, especially to prevent perturbation of non-target cortex regions. In order to achieve

this, we can study the design of a TMS coil capable of producing a maximum E-field in a

prescribed cortex volume, while maintaining the stimulation in other of non-target regions

of interest below a given threshold.

The problem of designing a TMS coil with minimum stored energy which produces a max-

imum electric field in a first target region of M points and minimizes the magnetic field in a

63


second region of interest of M ′ points can be posed as

max‖Eψ‖2

E′ψ

2 ≤ s

ψT Lψ≤ r

(7.3.6)

where r, s ∈ R+, and it has to be satisfied that r > s‖EC−1‖2

, otherwise the constraints are

redundant.

A similar problem can be equivalently posed by constraining the maximum value of the E-

field induced in the non-target region.

max‖Eψ‖2

E′ψ

∞ ≤ s

ψT Lψ≤ r

(7.3.7)

7.3.4 Optimised current

The proposed BEM formulation permits inclusion of the current density in the design process,

which can be used to increase the stimulator buildability. For instance, more practical wire

patterns can be achieved by controlling the minimum spacing between adjacent wires and

by spreading out some of the more closely packed windings. More precisely, it is known

that the l1-norm of the current density when minimised promotes sparsity of the windings,

whereas minimising the l∞-norm causes the wires to become more spread out, and evenly

so (27).

The problem of designing a minimum inductance TMS coil with sparse windings or increased

minimum wire spacing and capable of stimulating a prescribed region of interest formed with

64


M points can be then written as

max‖Eψ‖2

minψT Lψ+ ξ‖ j(ψ)‖p

(7.3.8)

where p = 1, 2 or∞. The weight ξ ∈ R is an user-definable regularisation parameter that

illustrates the trade-off between coil properties. As it is shown in Appendix A.1.2 a solution

of (7.3.10) can be found by solving

minψT Lψ+ξ‖ j(ψ)‖p

‖Eψ‖2(7.3.9)

7.3.5 Optimised temperature

The coil temperature is an important issue in the stability of the TMS system and patient

comfort. The use of coils with minimised maximum current ( section 7.3.4) can be a good

strategy to tackle this problem, as they exhibit significantly reduced peak temperatures (27).

Nonetheless, the presented IBEM formalism also allows the coil temperature to be incorpor-

ated and controlled in the designed process (9).

The problem of designing a minimum inductance TMS coil with optimised temperature and

capable of stimulating a prescribed region of interest formed with M points can be then

written as

max‖Eψ‖2

minψT Lψ+τ‖ΛC (ψ)‖p

(7.3.10)

where p = 1, 2 or∞. The weight τ ∈ R is an user-definable regularisation parameter that

illustrates the trade-off between coil properties. As it is shown in Appendix A.1.2 a solution

65


of (7.3.10) can be found by solving

minψT Lψ+τ‖ΛC (ψ)‖p

‖Eψ‖2(7.3.11)

66

CHAPTER

8TMS coil design: numerical results

The optimisation framework described in the previous chapter is as a powerful TMS coil

design approach. In this section it is used to design TMS stimulators with arbitrary geo-

metry, allowing the inclusion of new coil features in the design process, such as optimised

maximum current density or reduced temperature. Even the structural head properties have

been considered to produce more realistic TMS stimulators.

Several examples of TMS coils were designed and simulated to demonstrate the validity of

the proposed optimisation framework described in the previous chapter.

For sake of comparison, the approach used in Cobos Sanchez et al. (8) in which the op-

timisation is carried out by using classical techniques, such as simple partial derivation and

subsequent matrix inversion of the consequential system of linear equations, will be noted

as partial derivation optimisation (PDO)

Moreover,sinusoidal variation of the electric and magnetic fields ( f = 5kHz) has been ad-

67

8. TMS COIL DESIGN: NUMERICAL RESULTS

opted, and unless it is stated, an arbitrary coil current with peak value of 1 kA has been

considered. Additionally, depth and focality metrics were employed to describe the spatial

characteristics of the stimulation, where the human head has been modelled by a homogen-

eous sphere of 8.5 cm radius and isotropic conductivity. The cortical surface was assumed to

be at a depth of 1.5 cm from the surface of the head, so the cortex is described by a sphere

of 7.0 cm radius.

8.1 Minimum stored magnetic energy: Coil 1

We first design a TMS stimulator of a rectangular flat form of 14 cm × 7.5 cm located at

the x y−plane, and it is designed to maximise By in a prescribed region of interest (ROI),

which is made up of 400 points inside a spherical region of radius 2 cm that is centred on

the z-axis, and 4 cm below the conducting surface. In order to evaluated the optimality of

the coil obtained using the suggested approach, we compare it to equivalent one produced

using PDO.

Figure 8.1(a), shows the optimal nodal values of the stream function calculated using the sin-

gular vector formalism (in blue) with are different to those generated with the PDO approach

(in black). This discrepancy can be also found when analysing the wire-paths predicted by

both methods, and confirmed when evaluating the characteristics of each TMS coil in Table

1, where it can be seen that the TMS stimulator design provide a more efficient performance

than Coil 1 produced using PDO, which has higher values of L and R.

8.2 Full field maximized: Coil 2

Koponen et al. (24) have shown the design of minimum-energy TMS coils wound on a

spherical surface, this represents an interesting approach as this particular geometry is well

68

8.2 Full field maximized: Coil 2

0 500 1000 1500−1

−0.5

0

0.5

1

node number

ψ

PDO

Truly optimal

(a)

x (m)

y (

m)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

0.01

0.02

0.03

0.04

0.05

0.06

(b)

Figure 8.1: a) Optimal stream function values for a rectangular TMS coil described using a meshof 1522 nodes, designed with singular vector formalism (blue) and PDO (black). b) One quadrantof the wire paths of TMS coil 1 designed with singular vector formalism (blue) and PDO (black).

matched to the head.

Here we investigate the design of a minimum inductance (or equivalently minimum stored

energy) spherical TMS coil of radius 9 cm, which is designed to produce an optimised E-

field in a spherical ROI of 2 cm radius and centred 5 cm above the centre of the conducting

sphere.

The wire-paths of the solution to Coil 2 problem is shown in Fig. 8.2(a), where red wires

indicate reversed current flow with respect to blue. As expected, there is a higher density of

winding turns over the region of stimulation. The performance parameters of the spherical

TMS coil can be found in Table 8.1.

These wire arrangements can also be found in blue in Fig. 8.2(b) along with the solution

generated with the PDO approach (in black), where there is a clear shift in the wire pattern

between both coils.

69


(a)

φ (rad)

θ (

rad

)

−3 −2 −1 0 1 2 30

0.5

1

1.5

2

2.5

3

(b)

Figure 8.2: a) Wirepaths with 18 turns of Coil 2 designed with singular vector formalism. b) Thewire paths of TMS coil 2 designed with singular vector formalism (blue) and PDO (black).

8.3 Reduction of the undesired stimulation: Coil 3

The undesired stimulation in non-target cortex regions can be reduced by introducing a

second region of interest (ROI2), in which the E-field produced by the stimulator is minim-

ized. This strategy is evaluated by designing a minimum stored-energy spherical TMS coil of

radius 9 cm, constructed to produce an E-field which is both a maximum in a spherical ROI

and minimum in a second ROI2. Both volumes of interest are of 2 cm radius and formed by

400 points, where ROI2 is concentric to the conducting sphere and ROI is shifted by 5 cm in

the positive z-direction.

The coil is designed so the maximum value of the E-field in the second ROI2 is less than

half of that one produced by the corresponding TMS coil design using PDO, while the stored

energy cannot be more than 10% of the value of the PDO counterpart.

Coil 3 has been designed to mitigated the undesired stimulation in the prescribed non target

cortex region (ROI2), in order to evaluate this fact, the electric field induced in the ROI2

by Coil 3 is depicted in Fig. 8.4(a), which can be compared to that one generated by its

70

8.4 Optimised current: Coil 4

(a)

Figure 8.3: Wirepaths with 18 turns of Coil 3. Red wires indicate reversed current flow with respectto blue.

counterpart Coil 2 in Fig. 8.4(b). It can be seen how the values obtained for Coil 3 are lower

than those produced in the case of Coil 2, being the reduction of the electric field induced

greater than 70% at many points in the ROI2.

The price we have to pay for the mitigation of the undesired stimulation is a significant

reduction of the electric field induced in the target region ROI; and even the increase of the

E-field induced in other head parts.

8.4 Optimised current: Coil 4

By using the same rectangular planar surface as for coil 1, a slightly more complex coil design

problem can be posed by maximizing the magnitude of the E-field in the same the prescribed

spherical volume of interest (ROI) while minimizing the l∞ norm of the current density.

Figure 8.5(a) shows the wire path of the Coil 4, which is again concentrated over the region of

stimulation. It is formed by two lobes of eight turns, which have been produce by contouring

71


−0.02

0

0.02

−0.02

0

0.02−0.02

−0.01

0

0.01

0.02

x (m)y (m)

z (m

)

2

4

6

8

10

12

14

16

18

E(V/m)

(a)

−0.020

0.02

−0.02

0

0.02

−0.02

−0.01

0

0.01

0.02

x (m)y (m)

z (m

)

1

2

3

4

5

6

E(V/m)

(b)

Figure 8.4: Three-dimensional plot of the colour-coded modulus of the E-field (in V/m) at thesurface of the ROI2 induced by (a) Coil 2 and (b) Coil 3.

the optimal stream function with the same number of levels. It is worth noting the spreading

of wires fact which can be employed to increase the efficiency of the TMS stimulator by

allowing extra turns to be added. The relevant parameters for Coil 5 are detailed in Table

8.1.

8.5 Optimised temperature: Coil 5

By using the same rectangular planar surface as for coil 1, a slightly more complex coil

design problem can be posed by maximizing the magnitude of the E-field in the same the

prescribed spherical volume of interest (ROI), figure 8.1(a), while minimizing the l∞ norm

of the temperature.

Figure 8.5(a) shows the wire path of the Coil 4, which is again concentrated over the region

of stimulation. Again there is clear spreading of wires, fact which is expectable as regions

with maximum temperature coincide with those where the coil windings are closely spaced.

In fact, the temperature at the surface of Coil 1, Coil 4 and Coil 5 has been evaluated for the

72

8.5 Optimised temperature: Coil 5

x (m)

y (

m)

−0.02 0 0.02

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

(a)

x (m)

y (

m)

−0.02 0 0.02

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

(b)

Figure 8.5: a) a) Wirepaths with 18 turns of Coil 4. b) Wirepaths with 18 turns of Coil 5.

following thermal properties (9): heat transfer coefficient ht = 160 W m−2K−1 , the effective

thermal conductivity ke = 401 Wm−1K−1 and resistivity of the copper ρr = 168 Ω m.

For all cases, it can be appreciated that regions with maximum temperature coincide with

those where the coil windings are closely spaced, nonetheless coil 5 presents a lower tem-

peratures, and lower maximum hot spots.

73


x (m)

y (

m)

T* (K)

−0.02 0 0.02

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

5

10

15

20

(a)

x (m)

y (

m)

T* (K)

−0.02 0 0.02

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

5

10

15

20

(b)

x (m)

y (

m)

T* (K)

−0.02 0 0.02

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

5

10

15

20

(c)

Figure 8.6: The temperature distribution over the ambient plotted over the surface of the TMS a)coil 1, b) coil 4 and c) coil 5 .

L (µH) R(mΩ) D1/2 (cm) S1/2 (cm2) Ncontours

Coil 1 PDO 9.3 70 1.2 13.9 16Coil 1 7.2 58 1.2 12.7 16Coil 2 PDO 17.4 174 2.5 48.0 18Coil 2 12.6 156 2.4 46.9 18Coil 3 11.3 175 1.7 32.2 18Coil 4 10.1 47 1.2 14.3 a 16Coil 5 10.7 50 1.2 a 14.8 a 16

Table 8.1: TMS coil performance parameters: L is the inductance, R the resistance, D1/2 the effectivesurface area (focality), D1/2 the penetration or depth and Ncontours is the number of levels in whichthe stream-function is contoured to produce the wire paths. Simulated values of l and R wereobtained using FastHenry c© (23) using 1 mm diameter circular cross-section wire. In brackets arethe measured values L and R of the prototype Coil1.

74

APPENDIX

AMathematical foundations of

the physical model

A.1 Supporting vectors

Recall that the set of supporting vectors of a continuous linear operator between normed

spaces X and Y is defined as

suppv(T ) := arg max‖x‖=1

‖T (x)‖= x ∈ SX : ‖T (x)‖= ‖T‖.

The supporting vectors play a fundamental role in the geometry of Banach spaces due to fam-

ous classical results such as the Bishop-Phelps Theorem and the Hahn-Banach Theorem. We

refer the reader to (15) for a wide perspective on the previous theorems and generalizations

of them.

75

A. MATHEMATICAL FOUNDATIONS OF THE PHYSICAL MODEL

A.1.1 Generalized supporting vectors

The generalized supporting vectors appear in a implicit way in many optimization problems

in Physics and Engineering (see (25)). Here we will properly define them in a more general

and abstract setting.

Definition A.1.1. Let X and Y be normed spaces and consider a sequence (Tn)n∈N of continuous

linear operators between them. The generalized supporting vectors of (Tn)n∈N are defined as the

elements of

gsuppv ((Tn)n∈N) := arg max‖x‖2=1

∞∑

n=1

‖Ti(x)‖2.

Notice that it can easily happen that gsuppv ((Tn)n∈N) be empty. In order to avoid this, some

conditions are required on the sequence of operators and on the normed spaces such as

reflexivity for X and that (Tn)n∈N ∈ `2 (B(X , Y )).

We will focus now on the generalized supporting vectors of a eventually null sequence of

finite-rank operators A1, . . . , Ak ∈ Rm×n. For this we need to recall several linear algebra

basic concepts along with other basic concepts from the spectral theory of normed algebras,

such as the point spectrum of a continuous linear operator.

Throughout this section, the 2-norm of a matrix A ∈ Rm×n is considered to be the operator

norm of A between Rn and Rm when both are endowed with the Euclidean norm.

A matrix P ∈ Rn×n is said to be orthogonal provided that PT = P−1. Orthogonal matrices

induce isometries on `n2 := (R2,‖ · ‖2), that is,

‖P x‖22 = (P x)T (P x) = x T PT P x = x T x = ‖x‖22

for all x ∈ Rn. In particular, the 2-norm of an orthogonal matrix is 1.

76


It is well known that the eigenvalues of a symmetric real matrix are real. This fact remains

true in more general settings, like for instance in Operator Theory. If A∈ Rm×n is symmetric,

then λmax(A) stands for the largest eigenvalue of A and V (λmax(A)) := ker(A− λmax(A)I) is

the vector subspace of eigenvectors associated to λmax(A).

Before stating and proving the following crucial lemma, we would like to make the reader

beware that if D ∈ Rn×n is a diagonal matrix, then ‖D‖2 =

λmax(D)

.

From the spectral theory of C∗-algebras we can easily deduce the following lemma. However,

we will include its proof for the sake of completeness.

Lemma A.1.2. If A∈ Rn×n is positive semi-definite and symmetric, then

1. ‖A‖2 = λmax(A).

2. x T Ax ≤ λmax(A)‖x‖22 for all x ∈ Rn.

Proof. Since A is symmetric, we have that A is orthogonally diagonalizable, in other words,

there exists an orthogonal matrix P and a diagonal matrix D such that A = PT DP and the

eigenvalues of A are the elements of the main diagonal of D. On the other hand, since A is

also positive semi-definite, the eigenvalues of A are positive.

1. Since P and PT are both isometries, we have that

‖A‖2 = ‖PT DP‖2 = ‖D‖2 = λmax(D) = λmax(A).

2. By using again that P is an isometry and by relying on the above item,

x T Ax =

x T Ax

≤ ‖x T‖2‖A‖2‖x‖2 = λmax(A)‖x‖22.

77


We recall the reader that the set of smooth points of the unit ball of a normed space X is

defined as

smo(BX ) := x ∈ SX : ∃ x∗ ∈ SX ∗ with x∗(x) = 1 .

When smo(BX ) = SX we call X a smooth normed space. It is well known that all Hilbert

spaces are smooth.

Theorem A.1.3. Let A1, . . . , Ak ∈ Rm×n. Then

max‖x‖2=1

k∑

i=1

‖Ai x‖22 = λmax

k∑

i=1

ATi Ai

and

V

λmax

k∑

i=1

ATi Ai

∩ S`n2= arg max

‖x‖2=1

k∑

i=1

‖Ai x‖22.

Proof. First off, for any x ∈ Rn, and in virtue of Lemma A.1.2(2),

k∑

i=1

‖Ai x‖22 =k∑

i=1

x T ATi Ai x = x T

k∑

i=1

ATi Ai

x ≤ λmax

k∑

i=1

ATi Ai

‖x‖2.

Therefore

max‖x‖2=1

k∑

i=1

‖Ai x‖22 ≤ λmax

k∑

i=1

ATi Ai

.

Now, let w ∈ V

λmax

∑ki=1 AT

i Ai

∩ S`n2. Then

k∑

i=1

‖Aiw‖22 = wT

k∑

i=1

ATi Ai

w= λmax

k∑

i=1

ATi Ai

.

This shows that

max‖x‖2=1

k∑

i=1

‖Ai x‖22 = λmax

k∑

i=1

ATi Ai

78


and

V

λmax

k∑

i=1

ATi Ai

∩ S`n2⊆ arg max

‖x‖2=1

k∑

i=1

‖Ai x‖22.

Finally, let v ∈ arg max‖x‖2=1

k∑

i=1

‖Ai x‖22. One the one hand, in accordance with Lemma A.1.2(1)

we deduce that

∑ki=1 AT

i Ai

v

λmax

∑ki=1 AT

i Ai

2

≤

∑ki=1 AT

i Ai

2

λmax

∑ki=1 AT

i Ai

= 1.

On the other hand,

vT

∑ki=1 AT

i Ai

v

λmax

∑ki=1 AT

i Ai

=

∑ki=1 ‖Ai v‖22

λmax

∑ki=1 AT

i Ai

= 1,

which implies that

∑ki=1 AT

i Ai

v

λmax

∑ki=1 AT

i Ai

2

= 1.

The smoothness of `n2 allows us to deduce that

∑ki=1 AT

i Ai

v

λmax

∑ki=1 AT

i Ai

= v,

that is, k∑

i=1

ATi Ai

v = λmax

k∑

i=1

ATi Ai

v

and so v ∈ V

λmax

∑ki=1 AT

i Ai

∩ S`n2.

As a consequence, we easily obtain the well known formula of the 2-norm of a matrix.

Corollary A.1.4. If A∈ Rm×n, then ‖A‖2 =p

λmax(AT A) and V (λmax(AT A))∩S`n2= suppv(A).

79


A.1.2 Matrix norms

This first subsection is to show the equivalence of the following optimization problems. The

reader may notice that first one is a multi-objective problem and the last one is a convex

optimization problem.

max‖Aϕ‖

min‖ϕ‖⇔

max‖Aϕ‖

‖ϕ‖= 1⇔

min‖ϕ‖

‖Aϕ‖= ‖A‖

The following results throws some light on the problem described in Equation (7.3.3) in the

sense that logic conjugation of the two conditions of maximization and minimization makes

it an unsolvable problem.

Theorem A.1.5. Let A be an H × N matrix and consider a norm ‖ · ‖ in RN . Consider the

multi-objective optimization problem

max‖Aϕ‖

min‖ϕ‖ϕ ∈ RN

1. There does not exist ϕ ∈ RN such that, for all ψ ∈ RN , ‖Aϕ‖ ≥ ‖Aψ‖ and ‖ϕ‖ ≤ ‖ψ‖.

2. There are infinitely many ϕ ∈ RN such that, for all ψ ∈ RN , either ‖Aϕ‖ ≥ ‖Aψ‖ or

‖ϕ‖ ≤ ‖ψ‖. These solutions are the elements of the set

⋃

ε≥0

supp‖·‖(A,ε).

Proof.

1. Suppose to the contrary that there is ϕ ∈ RN such that ‖Aϕ‖ ≥ ‖Aψ‖ and ‖ϕ‖ ≤ ‖ψ‖

for all ψ ∈ RN . Since ‖ϕ‖ ≤ ‖ψ‖ for all ψ ∈ RN we must have that ϕ = 0 which then

80


contradicts that ‖Aϕ‖ ≥ ‖Aψ‖ for all ψ ∈ RN since Aϕ = 0.

2. In the first place, assume that there exists ϕ ∈ RN such that ‖Aϕ‖ ≥ ‖Aψ‖ or ‖ϕ‖ ≤

‖ψ‖ for all ψ ∈ RN . We will show that ϕ ∈ supp‖·‖(A,ε) for ε := ‖ϕ‖. Indeed, let

ψ ∈ RN such that ‖ψ‖< ε (= ‖ϕ‖). Then by assumption ‖Aϕ‖ ≥ ‖Aψ‖, which means

that ‖Aϕ‖ ≥ sup‖Aψ‖ : ‖ψ‖ < ε = max‖Aψ‖ : ‖ψ‖ ≤ ε ≥ ‖Aϕ‖ since the open

ball ψ ∈ RN : ‖ψ‖ < ε is dense in the closed ball ψ ∈ RN : ‖ψ‖ ≤ ε. As a

consequence, ϕ ∈ supp‖·‖(A,ε).

Conversely, we will show that every ϕ ∈ supp‖·‖(A,ε) verifies that ‖Aϕ‖ ≥ ‖Aψ‖ or

‖ϕ‖ ≤ ‖ψ‖ for all ψ ∈ RN . Indeed, fix any ψ ∈ RN . If ε ≤ ‖ψ‖, then we are done

because ‖ϕ‖= ε. If ε > ‖ψ‖, then we conclude that

‖Aϕ‖=max‖Aχ‖ : ‖χ‖ ≤ ε ≥ ‖Aψ‖.

What Theorem A.1.5(1) is saying is that if we decompose the previous multi-objective op-

timization problem into the logic conjunction of two single-objective optimization problems,

max‖Aϕ‖

min‖ϕ‖

ϕ ∈ RN

=

max‖Aϕ‖

ϕ ∈ RN∧

min‖ϕ‖

ϕ ∈ RN(A.1.1)

then there is no solution. Note that the first single-objective optimization problem in Equa-

tion (A.1.1) has no solution and the second one has a unique solution, which is 0. As a

consequence, the first way we proposed to interpret multi-objective optimization problems

is inappropriate here. The correct interpretation is the second way, whose solutions are given

in Theorem A.1.5(2).

81


Corollary A.1.6. Let A be an H×N matrix and consider a norm ‖·‖ in RN . The multi-objective

optimization problem

max‖Aϕ‖


is equivalent to any one of the following:

1. The optimization problem

max‖Aϕ‖

‖ϕ‖= 1ϕ ∈ RN

which consists of finding the elements of supp‖·‖(A), that is, the elements of RN at which

A attains its norm.

2. The convex optimization problem

min‖ϕ‖

‖Aϕ‖= ‖A‖ϕ ∈ RN

which again consists of finding the supporting vectors of A.

Proof.

1. In accordance to Theorem A.1.5, the solutions to the optimization problem

max‖Aϕ‖


are the elements of the sets⋃

ε≥0

supp‖·‖(A,ε).

Since supp‖·‖(A,ε) = εsupp‖·‖(A) for all ε > 0, it suffices to find the supporting vectors

82


of A, that is, the elements of supp‖·‖(A), which are precisely the solutions to the

max‖Aϕ‖

‖ϕ‖= 1ϕ ∈ RN

2. All we need to show is that the solutions to the convex optimization problem

min‖ϕ‖

‖Aϕ‖= ‖A‖ϕ ∈ RN

are the supporting vectors of A. Indeed, letϕ ∈ supp‖·‖(A). Ifψ ∈ RN and ‖Aψ‖= ‖A‖,

then we have that

‖ϕ‖= 1=‖Aψ‖‖A‖

≤‖A‖‖ψ‖‖A‖

= ‖ψ‖

which implies that ϕ is a solution of the convex minimization problem.

Conversely, let ϕ a solution of the convex minimization problem. We will prove that

ϕ ∈ supp‖·‖(A). By assumption, ‖Aϕ‖ = ‖A‖ so all we need to show is that ‖ϕ‖ = 1.

Indeed, it suffices to consider any ψ ∈ supp‖·‖(A). Since ‖Aψ‖ = ‖A‖ we have that

‖ϕ‖ ≤ ‖ψ‖= 1. The other inequality follows from the fact that ‖A‖= ‖Aϕ‖ ≤ ‖A‖‖ϕ‖.

A.1.3 The Cholesky decomposition ψT Lψ

In this subsection we will show the equivalence of the following optimization problems, the

last one of which is a convex optimization problem.

83


max‖Aψ‖2

minψT Lψ⇔

max‖Aψ‖2

minp

ψT Lψ⇔

max‖Aψ‖2

min‖Cψ‖22

⇔

max‖(AC−1)ϕ‖2

min‖ϕ‖22⇔

min‖ϕ‖2

‖(AC−1)ϕ‖2 = ‖AC−1‖2

Lemma A.1.7. If L is a N × N symmetric and positive-definite matrix, then there exists an

invertible N × N matrix C such that ψT Lψ= ‖Cψ‖22 for all ψ ∈ RN .

Proof. We will provide two different proofs:

1. Since L is symmetric and positive-definite, we can appeal to the Cholesky decomposi-

tion of L, that is, L = C T C . Indeed, notice that

ψT Lψ=ψT (C T C)ψ= (Cψ)T (Cψ) = ‖Cψ‖22 .

2. Consider the bilinear map (ϕ,ψ)L := ϕT Lψ. Since L is symmetric and positive-

definite, this bilinear map defines a scalar product which makes RN become a Hil-

bert space whose norm is ‖ϕ‖L := (ϕ,ϕ)12L =

p

ϕT Lϕ. Since all the Hilbert spaces

of the same dimension are linearly isometric, there exists a surjective linear isometry

T :

RN ,‖ · ‖L

→

RN ,‖ · ‖2

. If denote by C to the matrix associated to the isometry

T , then

‖Cψ‖22 = ‖T (ψ)‖22 = ‖ψ‖

2L =ψ

T Lψ

for all ψ ∈ RN .

Observe that the condition ψT Lψ = ‖Cψ‖22 for all ψ ∈ RN automatically implies that C is

invertible. Indeed, since C is a square matrix it suffices to show that is kernel is null. So

84

A.2 Applications to vector fields

let ψ ∈ ker(C). Then ψT Lψ = ‖Cψ‖22 = 0, which implies that ψ = 0 since L is positive-

definite.


A.2.1 Operators with null divergence

Fix N ∈ N and let bn : R3→ R3 be vector fields for every 1≤ n≤ N . Ifψ= (ψ1, . . . ,ψN )T ∈

RN , then the vector field

B :=N∑

n=1

ψn bn

verifies that

div(B) =N∑

n=1

ψndiv(bn).

Now fix H ∈ N with H < N and r1, . . . , rH ∈ R3. We can define now the following linear

operators Bx , By , Bz : RN → RH by

Bx(ψ) :=

N∑

n=1

ψn bnx(r1), . . . ,

N∑

n=1

ψn bnx(rH)

By(ψ) :=

N∑

n=1

ψn bny(r1), . . . ,

N∑

n=1

ψn bny(rH)

Bz(ψ) :=

N∑

n=1

ψn bnz (r1), . . . ,

N∑

n=1

ψn bnz (rH)

where bnx , bn

y , bnz are the components of the vector field bn for 1 ≤ n ≤ N . Notice that the

three linear operators above can be seen as H × N matrices.

85


We will find an example of an operator B such that div(B) = 0 and

arg max‖ψ‖2=1

‖Bxψ‖22 +

Byψ

22 + ‖Bzψ‖

22 ∩ arg max

‖ψ‖2=1‖Bxψ‖

22 +

Byψ

22 =∅.

Example A.2.1. Consider constant vector fields bn(r) := (bn1 , bn

2 , bn3) for all r ∈ R3 and 1 ≤

n≤ N. It is clear that div(bn) = 0 for 1≤ n≤ N and thus div(B) = 0. Observe that

Bx =

b11 b2

1 · · · bN1

b11 b2

1 · · · bN1

......

. . ....

b11 b2

1 · · · bN1

By =

b12 b2

2 · · · bN2

b12 b2

2 · · · bN2

......

. . ....

b12 b2

2 · · · bN2

Bz =

b13 b2

3 · · · bN3

b13 b2

3 · · · bN3

......

. . ....

b13 b2

3 · · · bN3

Now if we take bni = δin for i = 1,2 and bn

3 = 2δ3n 1≤ n≤ N, then

arg max‖ψ‖2=1

‖Bxψ‖22 +

Byψ

22 = arg max

‖ψ‖2=1ψ2

1 +ψ22

= S`N2∩ spane1, e2

86


and

arg max‖ψ‖2=1

‖Bxψ‖22 +

Byψ

22 + ‖Bzψ‖

22 = arg max

‖ψ‖2=1ψ2

1 +ψ22 + 4ψ2

3

= e3,−e3

in virtue of the fact that

ψ21 +ψ

22 + 4ψ2

3 = 1−

ψ24 + · · ·+ψ

2N

+ 3ψ23 < 4

if |ψ3|< 1.

A.2.2 The max scalar field associated to a vector field

Fix arbitrary points r1, . . . , rH ∈ R3 and consider a map E : r1, . . . , rH ×RN → R such that

E(ri , ·) is linear for all 1≤ i ≤ H. Define

B : RN → RH

φ 7→ B(φ) := (E(r1,φ), . . . , E(rH ,φ))

and

Emax : RN → R

φ 7→ Emax(φ) := ‖B(φ)‖∞ =max|E(ri ,φ)| : 1≤ i ≤ N

For t ∈ [0, 1] we let

87


Nt(φ) := i ∈ 1, . . . , N : |E(ri ,φ)| ≥ tEmax(φ)

dt(φ) :=max‖ri‖ : i ∈ Nt(φ)

Vt(φ) := ri : i ∈ Nt(φ)

St(φ) :=vol(Vt)dt(φ)

Theorem A.2.2. Fix arbitrary points r1, . . . , rH ∈ R3 and let E : r1, . . . , rH×RN → R be such

that E(ri , ·) is linear for all 1≤ i ≤ H. Let t ∈ [0, 1].

1. If t = 0, then dt(φ) =max‖ri‖ : 1≤ i ≤ N for all φ ∈ RN and thus dt is constant.

2. If dt is continuous, the it is constant.

3. Nt(λφ) = Nt(φ), dt(λφ) = dt(φ), Vt(λφ) = Vt(φ) and St(λφ) = St(φ) for all λ ∈ R

and all φ ∈ RN .

4. If φ ∈ RN is so that |E(ri ,φ)| 6= tEmax(φ) for all 1≤ i ≤ N, then dt is continuous at φ.

5. If φ ∈⋂

1≤i≤H ker (E(ri , ·)), then dt(φ) =max‖ri‖ : 1≤ i ≤ H.

6. If φ,ψ ∈ RN are so that αφ + βψ ∈⋂

1≤i≤H ker (E(ri , ·)) for some α,β ∈ R, then

dt(φ) = dt(ψ).

Proof.

1. Obvious.

2. Note that the range of dt is a finite Hausdorff topological space, thus it is discrete. On

the other hand, the domain of dt is RN , which is connected, therefore dt cannot be

88


continuous if the range of dt has more than one point.

3. We will show that Nt(λφ) = Nt(φ) for all λ ∈ R and all φ ∈ RN . The other ones are

similar. So, let λ ∈ R and all φ ∈ RN . The linearity of E(ri , ·) for all 1 ≤ i ≤ N allows

us to deduce the following chain of equalities:

Nt(λφ) = i ∈ 1, . . . , N : |E(ri ,λφ)| ≥ tEmax(λφ)

= i ∈ 1, . . . , N : |λ||E(ri ,φ)| ≥ |λ|tEmax(φ)

= i ∈ 1, . . . , N : |E(ri ,φ)| ≥ tEmax(φ)

= Nt(φ).

4. Write 1, . . . , N= I ∪ J where

I := i ∈ 1, . . . , N : |E(ri ,φ)|> tEmax(φ)

and

J := i ∈ 1, . . . , N : |E(ri ,φ)|< tEmax(φ).

Pick a sequence (φn)n∈N ⊂ RN converging to φ. We will show that (dt(φn))n∈N con-

verges to dt(φ). Notice that (E(ri ,φn))n∈N converges to E(ri ,φ) for all 1 ≤ i ≤ N

since E(ri , ·) is a linear map between real vector spaces of finite dimension (thus it is

continuous). In a similar way, (Emax(φn))n∈N converges to Emax(φ). Therefore, we

can find n0 ∈ N in such a way that for all n ≥ n0, |E(ri ,φn)| > tEmax(φ) if i ∈ I and

|E(ri ,φn)|< tEmax(φ) if i ∈ J . As a consequence, dt(φn) = dt(φ) for all n≥ n0.

5. If φ ∈⋂

1≤i≤H ker (E(ri , ·)), then |E(ri ,φ)| = 0 for all 1 ≤ i ≤ N , thus Emax(φ) = 0,

therefore Nt = 1, . . . , N, which implies that dt(φ) =max‖ri‖ : 1≤ i ≤ H.

89


6. In virtue of the third item of this lemma, note that

E(ri ,φ) = E(ri ,αφ) = E(ri ,−βψ) = E(ri ,ψ)

for all 1≤ i ≤ N and

Emax(φ) = Emax(αφ) = Emax(−βψ) = Emax(ψ)

therefore dt(φ) = dt(ψ).

Corollary A.2.3. Fix arbitrary points r1, . . . , rH ∈ R3 and let E : r1, . . . , rH × RN → R be

such that E(ri , ·) is linear for all 1≤ i ≤ H. If a > 0, then

arg maxd 1

2(x)≤a

‖B(x)‖∞ =∅.

Proof. If there exists a feasible element x ∈ RN with d 12(x)≤ a and B(x) 6= 0, then d 1

2(nx)≤

a for all n ∈ N in virtue of Theorem A.2.2(3) and (‖B(nx)‖∞)n∈N diverges to∞.

90

APPENDIX

BConclusions

In the first place, we would like to make the reader beware that Chapters 3, 4 and 5 can be

found in (18; 19; 20), Chapter 6 can be found in (21), Chapters 7 and 8 can be found in (6),

and Appendix A can be found in (7).

B.1 Ricceri’s Conjecture

With respect to Ricceri’s Conjecture, one of our main accomplishment is the following result

(see Theorem 5.1.12):

Theorem B.1.1. Every non-zero finite dimensional Hausdorff topological vector space enjoys

the anti-proximinal property.

We have actually approached the conjecture in the positive in Theorem 5.2.2:

91

B. CONCLUSIONS

Theorem B.1.2. There exists a non-complete real normed space enjoying the weak anti-proximinal

property.

Theorem 5.2.3 also constitutes a partial positive solution to Ricceri’s Conjecture:

Theorem B.1.3. If every totally anti-proximinal convex set containing 0 is quasi-absolutely

convex, then Ricceri’s Conjecture holds true.

In fact, if we assume that every totally anti-proximinal convex set has an absorbing translate,

then the following theorem (Theorem 5.2.4):

Theorem B.1.4. Let X be a Hausdorff locally convex topological vector space. If X is a Baire

space, then X satisfies the anti-proximinal property.

together with several classic results will give us the key to prove Ricceri’s Conjecture true.

Example B.1.5 (Positive Solution to Ricceri’s Conjecture). There exists a non-complete normed

space satisfying the anti-proximinal property. Indeed, it suffices to consider any non-complete

normed space which is a Baire space and apply Theorem 5.2.4. For an example of a non-complete

normed space which is Baire take a look at (3, Chapter 3) where it is observed that if E is a

separable, infinite-dimensional Banach space, then E contains a dense subspace M of countably

infinite co-dimension which is a Baire space.

Our final conclusion with respect to Ricceri’s Conjecture is that our positive approaches do

not leave enough room to believe in the existence of a counter-example. This is why in the

future we will gather our efforts to prove Ricceri’s Conjecture true.

B.2 TMS coils

The search for improved coil performances and more practical coils has prompted the need

to consider new electromagnetic properties.

92

B.2 TMS coils

An inverse boundary element method and efficient optimisation techniques were combined

to produce a versatile framework to design TMS coils. An extension to a previously presented

method has been described for redesigning gradient coils. This requires considerable user

input and results in suboptimal coils. Poole et al. developed a method that spreads the closest

wires automatically, but again the optimality of the resulting coils was not guaranteed.

An important issue in TMS is to determine the site and size of stimulated cortex, moreover

precise spatial localization of stimulation sites is the key of efficient magnetic stimulations.

Many problems in engineering require to determine the spatial distribution of electric cur-

rents flowing on a conductive surface, which must satisfy some given requirements for the

produced fields, electromagnetic energy, etc. The reconstruction of current distribution on

the conducting surface subjected to these constraints is an inverse problem, which when for-

mulated using boundary element methods can be posed as a convex optimisation. Here we

present a convex optimisation framework to tackle problems in Bioengineering, that permits

the prototyping of many different cost functions and constraints. Several examples of TMS

coils were designed and simulated to demonstrate the validity of the proposed approach.

The inverse boundary element method and generalised convex optimisation techniques were

combined to provide a more flexible framework to design gradient and shim coils. This new

method was used to design coils by minimising different types of norms of the current density.

Several examples of TMS coil were designed and simulated to demonstrate the suggested

method, as well as to investigate and elucidate the behaviour of some of the most important

TMS coil performance requirements.

93

APPENDIX

CPublications resulting from

this work

C.1 Off-starting publication

1. F. J. García-Pacheco: “An approach to a Ricceri’s conjecture”, Topol. Appl. 159 (2012),

3307–3313.

C.2 Ph.D. Candidate publications

1. F. J. García-Pacheco and J. R. Hill: “Advances on Ricceri’s most famous conjecture”,

Filomat 29 (2015), no. 4, 829–838.

95

C. PUBLICATIONS RESULTING FROM THIS WORK

2. F. J. García-Pacheco and J. R. Hill: “A partial positive solution to a conjecture of Ricceri”,

Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 57–67.

3. F. J. García-Pacheco and J. R. Hill: “Geometric characterizations of Hilbert spaces”,

Canadian Math. Bull. 59 (2016), 769–775.

4. C. Cobos Sanchez, F. J. García-Pacheco and J. R. Hill: “Computational framework for

the design of truly optimal TMS coils using an inverse boundary element method”, Int.

J. Numer. Meth. Biomed. Engng. (submitted).

C.3 Tangential publication

1. C. Cobos-Sanchez, F. J. García-Pacheco, S. Moreno-Pulido and S. Saez-Martinez: “Sup-

porting vectors of continuous linear operators”, Annals Funct. Anal. (submitted).

96

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