Payá, proximinalityand polytopes

David Yost

Workshop on Banach spaces Granada 2015

Mathematical research is a social activity.I have 8 Spanish co-authors. However 8 is a small number;Andreas Defant has 15.Rafa was my first.My Ph.D. was done in Edinburgh, then the home of NumericalRange.My thesis was about C ∗-algebras and proximinal subspaces.R. Paya-Albert. Numerical range of operators and structure inBanach spaces. Quart. J. Math. Oxford Ser. (2) 33 (1982), no.131, 357–364.andRafael Paya; Javier Perez; Angel Rodrıguez. NoncommutativeJordan C ∗-algebras. Manuscripta Math. 37 (1982), no. 1, 87–120.So...

M-ideals relate to both these topics.We recall:a subspace J of a Banach space is said to have the n-ball propertyin X if, whenever B1, . . . ,Bn are open balls in X , with

⋂ni=1 Bi

non-empty and J ∩ Bi non-empty for each i , then we also haveJ ∩

⋂ni=1 Bi non-empty.

This property is not a property of either J or X ; rather it describesthe way J sits inside X .It is well known now that the 3-ball property implies the n-ballproperty for all n, and this happens if and only if J0 is anL-summand in X ∗; such subspaces are called M-ideals. In aC ∗-algebra, the M-ideals are precisely the closed 2-sided ideals;there are many other natural examples.This topic began with Alfsen and Effros (1972).

The literature on M-ideals is now vast;here is all you need and more.

It can be shown that M has the n-ball property in X if and only ifit has the n-ball property in M ⊕Kx for every x ∈ X \M.To some extent, we can study the n-ball properties just in the caseof hyperplanes.A natural question is: does the 2-ball property imply the 3-ballproperty?The simplest counterexample is X = `1(3) andM = (x , y , z) : x + y + z = 0.The diagram shows why the 3-ball property fails.

Alfsen, Erik M.; Effros, Edward G.Structure in real Banach spaces. I, II. Ann. of Math. (2) 96 (1972), 98–128; ibid. (2) 96 (1972), 129–173.

Why does this example have the 2-ball property?A subspace M is proximinal if every point in X has a bestapproximant in M, i.e. if

PM(x) = m ∈ M : ‖x −m‖ = d(x ,M)

is non-empty, for every x . Holmes, Scranton and Ward (1975)showed that the 2-ball property implies proximinality, and that

PM(x)− PM(x) ≈ BM(0, 2d(x ,M)).

This is close to characterising the 2-ball property for hyperplanes.For simplicity, we will often consider only finite dimensional spacesfrom now on.

Theorem(Lima, 1977) A hyperplane M has the 2-ball property in X if andonly if there is a face F of the unit ball BX , parallel to M, so thatBX is the closed convex hull of −F ∪ F .

(⇒) We have M = ker f where ‖f ‖ = 1. Fix a ∈ X withf (a) = ‖a‖ = 1. Given x ∈ X with ‖x‖ < 1, the 2-ball propertygives us some

m ∈ M ∩ B(a, 1) ∩ B

(a− 2x

1 + f (x),

1− f (x)

1 + f (x)

).

Put

y = a−m and z =2

1− f (x)x − 1 + f (x)

1− f (x)y .

Then y ∈ F , z ∈ −F and x is a convex combination thereof.

In the finite dimensional case, M will have the 3-ball property ifonly if F is symmetric.More generally, the hyperplane of functions whose integral is zerohas the 2-ball property but not the 3-ball property in L1(µ) (exceptin the obvious trivial case).So, the best known examples of subspaces with the 2-ball propertybut not the 3-ball property are all hyperplanes. Of course,examples which are not hyperplanes can easily be constructed bytaking direct sums. Are there examples of codimension two whichdo not admit such trivial decompositions?

Banach Spaces and Classical Analysis

Kent, OhioJuly/August 1985

Alfsen and Effros noted that the 3-ball property is transitive (beingan L-summand is transitive).A convex body P is said to be reducible (Shephard, 1963) ifP = 1

2(Q −Q) for some convex body Q which is not a translate ofP (i.e. not symmetric).

Corollary

A Banach space has the 2-ball property but not the 3-ball propertyin some superspace, if, and only if, its unit ball is reducible, if, andonly if, it contains nontrivial sets of constant width.

Payá, Rafael; Yost, DavidThe two-ball property: transitivity and examples. Mathematika 35 (1988), no. 2, 190–197.

So there exist “non-trivial” examples of the 2-ball property.There is a 4-dimensional Banach space Z , containing noM-summands, but which contains a 2-dimensional propersemi-M-ideal, X , whose unit ball of X is an octagon. So X has noM-summands.D. Yost & R. Paya, The two-ball property: transitivity andexamples, Mathematika 35 (1988) 190-197.1987: visited Granada for a month, finished this paper, startedanother one.

Call a subspace absolutely proximinal there is a function f on R2

for which the identity

‖x‖ = f (d(0,PM(x)), d(x ,M))

holds. The case f (α, β) = α+ β was introduced by Godini (1983).J.F. Mena, R. Paya, A. Rodrıguez & D. Yost. Absolutelyproximinal subspaces of Banach spaces, J. Approx. Theory 65(1991) 46-72.

A subspace is Chebyshev if PM(x) is always a singleton. This led toA. Lima & D. Yost, Absolutely Chebyshev subspaces, Proc. CentreMath. Anal. Austral. Nat. Univ. 20 (1988) 116-127.Motivated by work of Kalton et al on twisted sums ofquasi-Banach spaces, it included

TheoremFor a fixed Banach space Y , the following are equivalent.(i) For every Banach space X , there is a Banach space Z ,containing X as a non-trivial L1-Chebyshev subspace, with Z/Xisometric to Y .(ii) The unit ball of Y ∗ is reducible.(iii) There is a homogeneous but non-linear mapping Ω : Y → Rsatisfying

|Ω(x) + Ω(y)− Ω(x + y)| ≤ ‖x‖+ ‖y‖ − ‖x + y‖.

The special case of the last result, when Y is finite dimensionaland its unit ball is a polytope, is:

TheoremLet P be a polytope which is symmetric about the origin, V itsvertex set, and X the ambient vector space. Then P is reducible if,and only if, there is a non-constant function ρ : V → X such that(i) for all adjacent vertices v and w, ρ(v)− ρ(w) is a multiple ofv − w,(ii) for all v ∈ V , ρ(−v) = ρ(v).

After years moving between Banach spaces, Lipschitzmultifunctions and decomposable convex bodies, I am now workingwith Guillermo Pineda and Julien Ugon in Ballarat, on lower boundtheorems for general polytopes.We denote by φ(v , d) the minimum possible number of edges, overall d-polytopes with v vertices.It is well known that φ(v , 3) is either 3v/2 or 1

2(3v + 1) dependingon the parity of v (Steinitz, 1906). Examples achieving thesebounds are easily constructed by successively slicing corners off atetrahedron or a pyramid.The 4-dimensional case was solved by Grunbaum in 1967. Heshowed that φ(6, 4) = 13, φ(7, 4) = 15, φ(10, 4) = 21, and thatφ(v , 4) = 2v for all other values of v .

We have solved the 5- and 6-dimensional cases of this question.

TheoremFor all odd v ≥ 6, φ(v , 5) = 1

2(5v + 3). For all even v ≥ 10, or ifv = 6, φ(v , 5) = 1

2(5v); φ(8, 5) = 22.

TheoremFor all v ≥ 7, except the following, φ(v , 6) = 3v;φ(8, 6) = 26, φ(9, 6) = 30, φ(10, 6) = 33, φ(11, 6) = 35,φ(13, 6) = 41, φ(14, 6) = 45, φ(18, 6) = 56 and φ(19, 6) = 59.

We know all values of φ(v , 7) except v = 17 and v = 22.

TheoremFix d. Then for all sufficiently large v,(i) φ(v , d) = 1

2dv, if either v or d is even (known).(ii) φ(v , d) = 1

2d(v + 1)− 1, if both v and d are odd (new).

What about low numbers of vertices?A simplex shows that φ(d + 1, d) =

(d+12

). A prism based on a

(d − 1)-dimensional simplex shows that φ(2d , d) = d2.Grunbaum proved that

φ(d + k , d) =

(d

2

)−(

k

2

)+ kd

for 1 ≤ k ≤ 4, and conjectured that it holds in fact for everyk ≤ d .We have shown that this is true, and moreover that the minimisingpolytope is unique.We also have precise values for φ(2d + 1, d) and φ(2d + 2, d), andcharacterisations of the minimising polytopes.

TheoremLet P be a d-dimensional polytope with d + k vertices, where0 < k ≤ d.(i) If P is (d − k)-fold pyramid over the k-dimensional prism basedon a simplex, then P has

(d2

)−(k2

)+ kd edges.

(ii) Otherwise the numbers of edges is >(d2

)−(k2

)+ kd.

(a) P2 = M(2, 0) (b) M(2, 1) (c) M(2, 2) (d) P3 (e) M(3, 1)

FIGURE 1. Multiplexes

(a) Pentasm3 (b) Pentasm4

FIGURE 2. Pentasms

Slicing one corner from the base of a square pyramid yields apolyhedron with 7 vertices and 6 faces, one of them a pentagon.We call this a pentasm.We will use the same name for the higher-dimensional version,obtained by slicing one corner from the quadrilateral base of a(d − 2)-fold pyramid. It has 2d + 1 vertices and can also berepresented as the Minkowski sum of a d-dimensional simplex, anda line segment which lies in the affine span of one 2-face but is notparallel to any edge.

TheoremLet P be a d-dimensional polytope with 2d + 1 vertices.(i) If P is d-dimensional pentasm, then P has d2 + d − 1 edges.(ii) Otherwise the numbers of edges is > d2 + d − 1, or P is thesum of two triangles.

This shows that the pentasm is the unique minimiser if d ≥ 5.If d = 4, the sum of two triangles has 9 vertices, and is the uniqueminimiser, with only 18 edges.If d = 3, the sum of two triangles can have 7, 8 or 9 vertices; theexample with v = 7 has 11 edges, the same as the pentasm.Summarising, φ(9, 4) = 18, and φ(2d + 1, d) = d2 + d − 1 for alld 6= 4.

Slicing one corner from the apex of a square pyramid yields apolyhedron combinatorially equivalent to the cube. Slicing onecorner from 3-prism yields a polyhedron combinatorially equivalentto the 5-wedge. Of all the polyhedra with 8 vertices, these are theonly two with 12 edges.

We show that for d 6= 4, 5, 7, analogues of these polyhedra are theonly minimisers of the number of edges, amongst polytopes with2d + 2 vertices.

Slicing one corner from the apex of a square pyramid yields apolyhedron combinatorially equivalent to the cube. Slicing onecorner from 3-prism yields a polyhedron combinatorially equivalentto the 5-wedge. Of all the polyhedra with 8 vertices, these are theonly two with 12 edges.We show that for d 6= 4, 5, 7, analogues of these polyhedra are theonly minimisers of the number of edges, amongst polytopes with2d + 2 vertices.

TheoremLet P be a d-dimensional polytope with 2d + 2 vertices.(i) If P is one of the two polytopes just described, then P hasd2 + 2d − 3 edges.(ii) If d = 3, d = 6 or d ≥ 8, and P is any other polytope, thenthe numbers of edges is > d2 + 2d − 3.

If d = 4, there are two more minimising polytopes with 10 verticesand 21 edges.If d = 5, the unique minimiser is the sum of a tetrahedron andtriangle; this clearly has 12 vertices and 30 edges; 30 < 32.If d = 7, there is a third minimising polytope with 16 vertices and60 edges.Summarising, φ(12, 5) = 30, and φ(2d + 2, d) = d2 + 2d − 3 forall d 6= 5.The case of 2d + 3 vertices appears to be difficult.

TheoremLet P be a d-dimensional polytope with 2d + 2 vertices.(i) If P is one of the two polytopes just described, then P hasd2 + 2d − 3 edges.(ii) If d = 3, d = 6 or d ≥ 8, and P is any other polytope, thenthe numbers of edges is > d2 + 2d − 3.

If d = 4, there are two more minimising polytopes with 10 verticesand 21 edges.If d = 5, the unique minimiser is the sum of a tetrahedron andtriangle; this clearly has 12 vertices and 30 edges; 30 < 32.If d = 7, there is a third minimising polytope with 16 vertices and60 edges.Summarising, φ(12, 5) = 30, and φ(2d + 2, d) = d2 + 2d − 3 forall d 6= 5.The case of 2d + 3 vertices appears to be difficult.

It is well known that there is no polyhedron with 7 edges. Moregenerally a d-polytope cannot have between 1

2(d2 + d − 2) and12(d2 + 3d − 4) vertices, inclusive.Grunbaum [p 188] discusses gaps in the possible number of edges,pointing that a second gap opens when d = 6 and a third gapopens when d = 11. Our main theorem shows that there areinfinitely many gaps.More precisely, in dimension n2 + 2, there is no polytope with12(n4 + 2n3 + 4n2 + 3n + 4) edges.The cyclic polytope C (n2 + n + 2, n2 + 2) has one edge less, andthe free join of an (n2 − n)-dimensional simplex and an(n + 1)-prism has one edge more.For example: a 27-dimensional polytope cannot have 497 edges.But there is a cyclic polytope with 496 edges, and a multiplex with498 edges.

Theorem(i) Fix n ≥ 4. For any d ≥ n2, there is no d-polytope with between φ(d + n + 1, d) + 1 and φ(d + n + 1, d) + n − 3 edges.(ii) Fix n ≥ 5. If d = n2 − j, where 1 ≤ j ≤ n − 4, then there is no d-polytope with between φ(d + n + 1, d) + j + 1 and

φ(d + n + 1, d) + n − 3 edges.(iii) Fix n ≥ 1. In dimension d = n2 + j , where j ≥ 2, there is nopolytope with between

(d+n2

)+ 1 and

(d+n2

)+ j − 1 edges.

Vamos a transformar unas canas in teoremas.