19_1.tifDiscrete Comput Geom 4:19-40 (1989) C Diserete &
C~nnltmtatk~tal
, eOmeot
Finite Sphere Packing and Sphere Covering*
G. Fejes T6th, ~ P. Gr i t zmann , 2 and J. M. Wills 2
1 Mathematical Institute of the Hungarian Academy of Sciences,
Refiltanoda u. 13-15, H-1053 Budapest, Hungary
2 Mathematical Institute of the University of Siegen,
Hoelderlinstrasse 3, D-5900 Siegen, Federal Republic of
Germany
Abstract. A basic problem of finite packing and covering is to
determine, for a given number of k unit balls in Euclidean d-space
E d, (1) the minimal volume of all convex bodies into which the k
balls can be packed and (2) the maximal volume of all convex bodies
which can be covered by the k balls. In the sausage conjectures by
L. Fejes T6th and J. M. Wills it is conjectured that, for all d
-> 5, linear arrange- ments of the k balls are best possible. In
the paper several partial results are given to support both
conjectures. Furthermore, some relations between finite and
infinite (space) packing and covering are investigated.
1. Introduction
Let y[d deno te the set o f convex bod ies in Euc l idean d - space
E d, i.e., the set of all compac t convex subsets of E d with n o n
e m p t y interior . Fur ther , let B d be the unit ball in E d. A
bas ic p r o b l e m of the theory of finite packing is to de te
rmine for a given posi t ive in teger k the minimum of the volume
of all convex bodies into which k translates of B d can be
packed.
In the p lane this p r o b l e m and several ramif ica t ions have
been extensively studied. In par t icu la r , Fe jes T6th [5] (see
also [6]) showed that finite packings with B 2 canno t be dense r
than an op t ima l pack ing o f d isks in the whole p lane , and
Wills [25] con jec tu red that this also ho lds (except for some
smal l k) in E 3 and E 4. Fur ther , a cco rd ing to G r o e m e r
[11] and Wegner [22] ex t remal finite
~ p a p e r was written while the first named author was visiting
the "Forschungsinstitut fiir Geistes- und Sozialwissenschaften" at
the University of Siegen.
20 G. Fejes T6th, P. Grilzmann, and J, M+ Wills
packings of B ~ are essentially hexagonal parts of the densest
lattice packing of disks in E". This result has no direct analogy
in E ~ and E + (except perhaps for large k). in the optimal
arrangement of three balls in /:.'+ the centers are not, as one
might possibly expect, the vertices of a regular triangle, but
rather they lie on a straight line. Such linear arrangements are
called sausages, the name originating from the shape of the convex
hull of the balls, in E ' sausages seem to be best possible up to
56 balls (see Wills [23], [24]). Then the extremal configurations
change drastically and clusters of balls yield greater densities,
in E + the same phenomenon occurs, but much later, perhaps between
50,000 and 100,000 balls (see [23] and [24]), which justifies the
name sausage catastrophe.
In E: the situation changes completely. In E J, d ~ 5, sausages
seem to be best possible for any number of k unit balls. This is
Fejes T6th's [7] sausage conjecture for finite packings.
Correspondingly, a basic problem of the rheas" ¢~Jinite covering is
to determine, for a given positive integer k, the maximum o / the
volume of all conL~e.~ bodie.s which can be covered by k translates
of B '~. in this context there seems to be quite a strong analogy
between finite packings and finite coverings with the unit ball. in
particular, as a counterpart to Fejes T6th's sausage conjecture,
Wills [24] conjectured that sausage arrangements are also extremal,
for d ~ 5, in the case of finite coverings of B d.
In spite of their importance it has not yet been possible to prove
either of these two sausage conjectures completely in any
dimension. But there are a number of results which support the
conjectures. For a survey of the results known so far, refer to
Gritzmann and Wills [ 10].
In the present paper we give further evidence to support both
conjectures. Furthermore, we point out some relations to packing
and covering with B d with respect to the whole space. Finally, we
present a simple result concerning sausage catastrophes and outline
some possible applications in chemistry.
In Section 2 we give some basic notation. In Section 3 we introduce
a uniform concept of packing and covering densities which, in
particular, makes the occur- rence of sausage and sausage
catastrophe phenomena much more lucid. Section 4 contains the
statement of our results, the proofs of which are the content of
Sections 5, 7, 9, and 10. In Sections 6 and 8 we give upper and
lower estimates for the volume of parallel bodies of the regular
simplex which are essential for the proofs of Theorems 3 and 4, and
which may be of interest themselves.
2. Basic Notation and the Sausage Conjectures
Let V = V a denote the usual d-volume of convex bodies and, in
particular, set
(od = V( B '~ ) = TI. d12
I'( 1 + d/2)"
If K is an arbitrary convex body and if r runs through all
nonnegative real numbers then, by Steiner's formula, V ( K + rB d)
is a polynomial in r of degree
Finile Sphere Packing and Sphere ('o~ering 21
d which can be written in the form d
V(K +rBat= V wa ,V,(K)r a ' J (i
(see. e.g.. Hadwiger [ 12])+ The coefficients Vo(K }, .+. , Ve(K )
are the intrinsic volumes of K (see McMullen [16])+ Let us remark
that the intrinsic volumes are just a renormalization of the
quermass integrals. Therefore, regarded as func- tionals on L//e
they have the well+known properties of the quermass integrals, but,
moreover, they are independent of the dimension o f the space in
which K is embedded. Of particular service is the following
description o f the intrinsic volumes V, o f polytopcs. Let P be a
polytope and let F , (P) denote the set of all /-dimensional faces
f of P. Furthermore, let C ( f ) be the cone with vertex 0 of outer
normais of P taken at any relatively interior point o f f and let
a(.f l be the fraction o f the linear hull of (,+(+f) taken up by
C( f ) . Clearly+ 0 < a ( f ) -~ 1. But, m o r e o v e r ,
K~ P~ = v v'(.flc,( f ) .
Let us stress the fact that the sets f + C(.f) dissect E a+ and so,
in particular, they dissect P + B a+
Let k c I~1 and let
. i t : {(2i, 0 . . . . , 0 ) + Be[i= ! . . . . . k}.
Furthermore, let 11~ be a set of k translates of B a, the centers
of which are equally spaced on a straight line at such a distance
that, for all such configurations. +11"~ covers a convex body of
greatest volume.
Then the sausage conjectures can be expressed as follows:
Sausage conjecture .[or finite packings For d ~ 5 V(conv( :~) ) is
the minimum of the volume of all convex bodies into which k
translates of B d can be packed.
Sausage conjecture for finite coverings For d ~ 5 ~¢~ contains a
convex body o f maximal volume of all such convex bodies that can
be covered by k translates of B a.
We remark that there is an extremely useful alternative statement o
f the sausage conjecture for finite packings in terms of parallel
bodies.
Let C~ denote the convex hull of the centres of k nonoverlapping
translates of B ~ and let S~ be a segment of length 2 ( k - !),
Then the conjecture is that
V ( S k + B ' ~ ) < _ V ( G + B d) for d :~5.
Clearly, this way o f stating the conjecture is well suited for
methods involving
intrinsic volumes. If A is a discrete set of points in E a and a c
A then the set of all points of
E a which are not farther away from a than from any other point of
A is called the Voronoi polyhedron of a. Departing from the usual
notation, for a fixed convex body K we call the intersection o f
all Voronoi polyhedra o f points o f A c~ K with
K Dirichlet ceils with respect to K+
22
The rest of the notation used below Hadwiger [12] or Rogers
[20]).
G. Fejes T6th, P. Gritzmann, and J. M. Wills
is quite standard (see, for example,
3. Packing and Covering Densities
Let A = {a~, a2 . . . . } be a finite or infinite discrete set of
points in E d. We first introduce the so-called rt-density and
T-density of the system ~A = [ a , + B a, a2+ Ba,. . • ] of
translates of B a. (The restriction to the case of the unit ball is
merely a matter of simplicity and clarity. In fact, with some
obvious changes most of the things discussed in this section hold
for arbitrary convex bodies.) We use C d to denote the unit cube
with its edges parallel to the coordinate axis and with center O.
We write, for A e R, A > O,
"/T(~A, ,~) = max[rod c a r d ( A n AC d)/ V(AC d n C)IC e ~d
^ ( A n ^ c a ) + B d c C],
T(~A, A) = min[tod card(A n AC d ) /V (AC d n C)I C e ~fa
A C c ( A n A C d ) + Bd].
Then the rr-density and the T-density of ~A is defined as
follows:
rr(~A) = lim sup(~A, h ) A ~ o O
T(~A) = lim inf T(~A, h). A ~ o O
The system ~A is called a packing if each pair of translates ai + B
d, aj + B d ( i # j ) are nonoverlapping, i.e., if they do not have
interior points in common. Clearly, if ~3A is a packing then ~r(~A)
--< 1. Furthermore, T(~A) --> 1 in general.
We use the 7r- and y-densities to define packing and covering
densities for all the problems considered below, including the
classical densities for packings and coverings with respect to the
whole space (see, e.g., [20]), the densities of finite packings and
coverings (see e.g., [10]), and also some intermediate types.
Let us start by defining certain types of packings and coverings.
Let 1 -< n <- d. Then ~A is called an n-dimensional packing
of B d if ~A is a packing and A is contained in some n-dimensional
affine subspace E of E d, usually identified with E ~ such that
conv (A)= E. ~A is called an n-dimensional covering with B d if A c
E" such that E" c A + B d. Obviously, d-dimensional packings and
coverings are the classical packings and coverings with respect to
the whole space E a. I f A is finite and ~A is a packing, then ~A
is called a finite packing. I f A is finite, C e ~a, and C c A + B
d, then ~3A is called a finite covering of C. If, further, k e 1%1
and ~A is a finite packing or covering with ca rd (A)= k, then ~A
is called a k-packing, k-covering, respectively.
Finite Sphere Packing and Sphere Covering 23
Now we associate with B ~ the following densities:
6k(B d) = sup[~(~A) I ~A is a k-packing of Bd],
Ok(B d) = in f [y (~a ) l ~A is a k-covering with Bd],
6"(B e) = sup[rr(Y~A)i ~A is an n-dimensional packing of
Bali,
O"(B d) = inf[y(~3a) l ~ a is an n-dimensional covering with B d
].
As usual, we further set ,5(Bd)=6e(Bd), o(Bd)=od(Bd) . The reason
for introducing the subdensities 6", O" is precisely that the
sausage and sausage- catastrophe phenomena become much more evident
by means of a study of these densities.
To start with let us consider the packing problem. Since the
minimal convex set which contains a given set is its convex hull
it
is easy to see how the densities of n-dimensional packings of B e
are related. In fact, we have
8,,(Bj ) tog 6(B"). tontod-n
Since the gamma function F(x) is strictly convex the ratio tod
(O')nO)d-n)-! is also strictly convex (considered as a discrete
function of n, 1 ~ n ~ d). To give an impression of the size of
this ratio let us remark that
to,toe-, \ n / 2 / for even d, n,
and
77" < gl(Ba).
6(BJ) -< o'a,
where O'd denotes the ratio of the sum of the volumes of the
intersection of d + 1 unit balls centered at the vertices of a
regular simplex of side 2 to the volume of the simplex. Since this
bound is less than Blichfeldt's bound [3]
<a+2(1C2
24 G. Fejes T6th, P. Gritzmann, and J. M. Wills
which, for d - 7 , is less than 8~(B 'l) we have, for
d>-7,
8(Bd)--< o',~ < 8~(O~).
Accordingly, since the sequence
is convex in n for fixed d, we have
S" (B ' t )<-81(B '*) for 1-<-n<--d, d>-7.
But using the exact values of o'd (see, e.g., Leech [15]) it is
easy to verify that this inequality is valid for d > 5.
Figure 1 illustrates the behavior of the subdensities in dimension
24. For the calculations we used the densest known packings of B d.
Further, Figure 2 shows the subdensities for d -< 10 again with
the uncertainty of having used the best- known lattice packings for
d = 9, 10 since in these dimensions the exact quantities have not
been determined as yet. Roughly speaking, the figures show that,
for small n, ~( B a ) < ~" ( B a ) with the extreme case of
one-dimensional arrangements.
So the sausage conjecture for finite packings may be regarded as a
generaliz- ation of this fact to the case of finite packings of B
a.
Now let us turn to coverings. In contrast to packings, in order to
determine the subdensities O"(B a) we have to solve a simple
extremum problem to calculate the extra dilatation factor to be
used to maximize the contained convex body asymptotically. Doing
so, we obtain
d a/2 lo d
2 4 6 8
SUBDENSITIES for d=24
PACKING DENSITIES
Fig. 2
which is, for large d, d - n, by Stirling's formula,
,O.(Bd)~v/~/n(d-d-n)O(B").
d e'--~e Tn--O(B a)
(where rd denotes the ratio of the sum of the volumes of the
intersection of a regular simplex of side [2(d + 1)d- l ] ~/2 with
d + 1 unit balls centered at its vertices to the volume of the
simplex) it is easy to see that, for coverings with B d,
subdensities behave like packings of B d, thus motivating the
sausage conjecture for finite coverings.
To close Section 3 let us give one further definition for packings.
Let ka denote the largest number k such that
8k(a d) = 7r(~k).
If such a number does not exist we set kd = o0. kd is called the
sausage-catastrophe
26 G. Fcjes T6th, P. Gritzmann, and J. M. Wills
number of finite sphere packings. (Clearly, the sausage conjecture
for packings states that kd = oo for d --> 5 in this notation.)
For reasons that will become clear in the next section, for a
d-dimensional packing lattice ~3 of B a let k(@) denote the largest
number k such that
whenever A c ~d, card(A) = k Let us finally remark that such notion
also makes sense for finite coverings
(see [24]).
4. Results
We show that the sausage conjectures hold for several types of
arrangements. Let us start with a counterpart to the results of
Betke et al. [2] and Betke and Gritzmann [1] who showed that Fejes
T6th's sausage conjecture holds if the dimension of Ck is
sufficiently small compared with d, i.e., it holds for "flat"
arrangements. Our first theorem gives a corresponding result for
"large" arrange- ments.
Theorem 1. Let A c E d, card(A) = k be such that ~A is a packing.
Further, let Ck = conv(A), t r eR , t r< ~r(~A). Then
d X (~--~d2d--')Od--iE(Ck) --<0.
i=0
Setting, in particular, t r= ~r(3~k) Theorem 1 becomes a sausage
criterion showing that a certain class of finite packings of B d
yields a lower density than the sausage arrangements. In
particular, Theorem 1 implies that, for counter- examples to the
sausage conjecture, the insphere radius of Ck must be small
compared with the circumsphere radius.
Corollary. Let A c E d, card(A)= k be such that ~A is a packing.
Further, let Ck = cony(A) and let r, R denote the insphere radius
and circumsphere radius of Ck, respectively. If, moreover, 7r( ~k)
<- ~'( ~A), then
l + r < x/2 ( l + R ) [ ( 2 ~r )-t/Z( d + 2 )( d + l ) t/2] t/
a.
Let us point out that for large d this inequality is asymptotically
of the form
R r < - ~ + x / 2 - 1 .
(For coverings a result of the same spirit was proved by Gritzmann
and Wills [9].)
Finite Sphere Packing and Sphere Covering 27
Our second theorem is closely related to Theorem 1 in the sense
that it again deals with " large" arrangements of closely packed
balls, the so-called kissing number configurations. By a kissing
number configuration we understand a packing ~A of B d such that
there is an element aoC A so that I lao-al l = 2 for all a e A\{ao}
and, further, if there is an x c R a with Ilao-xll = 2 and I l a -
x ll-> 2 for all a ~ A\{ao} then x e A. So, intuitively
speaking, a kissing number configur- ation is a maximal packing of
B a such that each ball touches a given one. In a sense, kissing
number configurations may be regarded as building blocks of densest
d-dimensional packings of B d. However, Theorem 2 shows that, at
least for large d, they do not serve as counterexamples for Fejes
T6th 's sausage conjecture.
Theorem 2. Let d >- 12 and let ~A be a kissing number
configuration in E d o f cardinality k. Then
Let us remark that this result is not true for arbitrary
dimensions. In fact, in the plane the kissing number configuration
of seven disks has greater zr-density than the respective sausage
arrangement. But, as a special case of the sausage conjecture for
packings, Theorem 2 should hold for d -> 5.
Now we turn to special arrangements which seem to play a crucial
role in the understanding of Euclidean d-space manifesting itself
not only in packing and covering problems but also in different
problems such as determining the lattice point enumerator of convex
bodies. These are configurations related to the regular simplex T
d.
Theorem 3.
(a) Let d >-5, n ~ N, i <-d. Further, let A be the set o f
vertices of a regular i-dimensional simplex of side 2 in E d.
Then
"ff ( ~ A ) <~ 'D'( ~:;i+ 1)'
(b) Let s ~ R, s >- 2. Further, let A~ be the set o f vertices o
f a regular d-dimensional simplex o f side s. Then, as d ~ oo, we
have
7 r ( ~ , ) - > 2 -'/~- ~°~ ~ for s<-2#-;, ,/-d
7r(~A$):>2 -a/2 for s<-- - . 13
Part (a) o f the theorem shows that the ~r-density of the simplex
configuration is tess than the respective sausage density, at least
for d > 5. In fact, the same result is true even for d -> 3
(and fails for d = 2). The reason for the assumption d _~ 5 is that
we prefer a very short proof in the spirit of Section 3 rather than
a longer and much more technical one.
28 G Fejes T6th. P. Gritzmann. and J. M Wills
Part (b) of the theorem proves that, on the other hand, the
~v-density of the simplex configuration is much greater than the
density 8(B d) of d-dimensional sphere packing. In fact, by
Kabatjanski and Levenstein [ 14],
~(B # ) ~ 2 -o ~,*, ~ ,
Surprisingly we even obtain a finite packing of greater density
than 8( B ~ ) if we arrange d + I balls such that their centres
form a regular simplex of side ,/d/13. To demonstrate what this
means, in dimension, say, d = 169,676,676, the distance of the
balls may be 1000.
Let us now turn to asymptotic results for finite sphere coverings.
As a counter- part to part (a) of Theorem 3 we show
Theerem 4. Let d ~ 3, So = (2/(d - I)) */~. Further, let Ao be the
set of vertices of a regular d-dimensional simplex of side so. Then
we have
d + l
( in d ) '~'"
Compared with Coxeter el aL's [4] asymptotic bound d/ev'~ the
theorem shows that at least for high dimensions the y.density of a
regular simplex of suitable side is less than O(Bd). For coverings,
this is, indeed, a result similar to Theorem 3(b) for packings. It
would, of course, be nice to carry over part (a) as well. The
problem is that contrary to finite packings, where the least convex
body that contains all balls of a given arrangement is simply the
convex hull, convex bodies of greatest volume covered by a given
arrangement are not characterized. So, for finite coverings we have
one more optimization process (see Section 3).
Let us further point out that for a certain ramification of the
finite covering problem considered in [9], the so-called
cocoverings, the density of the simplex arrangement is much worse
than O(B d).
Now let us finally turn to the phenomenon of sausage catastrophes
for packings of B 3. This problem is of particular interest because
of its connections to chemistry.
As it is well known, the study of densely arranged configurations
of three- dimensional balls gives some insight into the behavior of
solids and liquids. For example, the molecular properties of many
crystals which, of course, bear a lattice structure can be
described, at least approximately, as the effect of forces on a
huge number of closely packed balls. In view of this close
connection it is not very surprising that there is also a
phenomenon in physical chemistry that corresponds to our sausage
catastrophes in E 3.
In fact, such phenomena of one-dimensional growth of atomic
structures up to a critical limit were observed by chemists and
engineers almost 40 years ago. Such metal threads, the so-called
Whiskers, caused short circuits in condensers. In particular, the
case of iron whiskers is o f some interest because of their
extremal stretching properties. Since the iron atoms are usually
arranged in the space. centered cubical lattice our last result
(which should serve as an illustrating example) deals with the
sausage catastrophe for the space-centered cubical lattice.
Finite Sphere Packing and Sphere Covering 29
Theorem 5. Let u = 2/v[3 and let ~ be the lattice generated by (2u,
0,0) , (0, 2u, O), (u, u, u). Then
k(~3) <- 23,968.
5. Proof of Theorem I, Its Corollary, aml Theorem 2
According to an estimate of Rogers [ 19] we have, for every Voronoi
polyhedron of a d-dimensional packing of B d,
gO d v ( P ) ~ ~ .
or,,
The following lemma shows that in some sense we can also make use
of this inequality in the case of finite packings of B a.
Lemma I. Let A c E d. card(A) = k such that 9Ba is a packing and Ck
= cony(A). Then there is a finite set A" with A c A', 3~ A. being a
packing such that the following property holds. Let a c A and let
Po be the Voronoi polyhedron of a with respect to A'. Then
P o c C k + 2 B d.
Proof. Let C = Ck + B d. Now we successively add points of bd (C)
to A such that the distance of any two points is at least 2. After
finitely many steps we obtain a maximal set A'. Clearly, ~ a is a
packing. Then i n t ( P o ) n b d ( C ) = O , which yields the
assertion. E3
Proof of Theorem !. Let A = { a, . . . . , ak } and let D, . . . .
, Dk denote the Dirichlet cells of a z , . . . , ah with respect to
Ck + 2B d. Further, let PI . . . . . Pk be respective Voronoi
polyhedra according to Lemma 1. Then we can apply Rogers's estimate
to P, . . . . . P,. Thus
t )-I
c r V ( G + B d ) ~ k w d = k g o d ~. V(D~) V(C~+2B d)
)-' l..oa V(P~) V(Ck +2B d )
, I
d d cr E god-,V~(Ck)~O'd Y- ca,~-,2d-'V~(C*)
I - - I i ' l
30
'
V~ are
we have
1 + d
Thus, by Theorem 1
~ , / d l + l (l + r)a <d + 2 2a/2( l +R)
which yields the assertion. []
Proof of Theorem 2. Let ~A be a kissing number configuration of
cardinality k in E d. Assume that 0= ao~ A and all other balls of ~
a touch B d. Let r be the maximum radius such that rB a c Ck =
conv(A). We consider a point x ~ rB d n bd(Ck) and a supporting
hyperplane H of Ck through x. Since we cannot add another ball to
~A without violating the packing or kissing number property, the
radius of 2B d ca H is at most v~ and we have r - 1. It follows
that
2dtOd <-- V(Ck + Bd).
Let Ma denote the maximum number of nonoverlapping equal spheres in
E d
that can touch a sphere of the same size. In other words, Ma + I is
the maximum cardinality of a kissing number configuration in E d.
Since
V( Sk + B d) <-- V( S ~ + B d) < 2Matod_l,
Theorem 2 will be proved by showing that, for d -> 12,
Ma <_ 2a_ t rod ('Od - 1"
Finite Sphere Packing and Sphere Covering 31
A result of Rankin [18] states that
Me<- 2-3/2 old_2
old-3 (sin t)d-2(COS t--COS "/4) dt dO
To estimate the upper bound, observe that iterative integration by
parts yields, for arbitrary nonnegative integer m,
thus
fo ~/4 1 "1 ( m-' d + 2j 1 ) sind-2 tdt=(d-1)2d/2i~=o\~=o d + 2 j +
1
0o( d+2j ) f"/4sina+2"ttdt +~ d + 2 j - 1 ao
1 m (__d_d ~ ' 1 -> (d -1 )2 d/2 ~ o k d + l ] 2 ~'
i: /4 d + 1 1 sin d-2 t dt >-- (d - 1)(d + 2) 2 (d-2)/2"
Using this inequality we deduce
Io '~/4 1 1 sin d-2 t(cos t - c o s ~ /4) dt ~- (d - 1)(d + 2) 2
~a-1)/2
and so, by means of Rankin's estimate,
M a < d ( d + 2 ) 2 t d - 4 ) / 2 t°a . old-1
Thus we have
at least for d -> 14. But applying the estimate
~ / 217" OI d d + 1 Ola-~
and using the upper bounds
M12<- 1416, MI3 <- 2233,
32 G. Fejes T 6 t h , P. G r i t z m a n n , a n d J, M.
Wills
which are due to Odlyzko and Sloane [17], it is obvious that the
inequality holds for d -> 12 which completes the proof of
Theorem 2. []
6. Upper Bounds for the Volume of Parallel Bodies of the Regular
Simplex
For the proof of Theorem 3 we shall need some upper estimates of
the volume of parallel bodies of the regular simplex. Let T d be
the regular d-dimensional simplex of side 1 and s e R, s -> 0.
Then, by Steiner's formula,
d d "~- CA) i V(sTd+Bd) = ~, t°d-,s'Vi(ra) E d-,S E a ( f ) V i ( f
) •
i=0 i=0 f ~ F t ( T d )
d+ 1] /-dimensional faces, Since T d has \ i+ 1 ]
( i+1 ) I/2 simplices of volume i!2 ~ / ~ ' we have
all being regular /-dimensional
~. ( d + l ] ( i+l) ' /z V(sTd + Ba)=~=o \ i+1 ] i!2 i/-----5-
t°a-iaisi'
where a~ denotes the external angle a(T ~) of T i considered as a
face of T d. In particular,
Let
Then
1 etO=d+ 1, a a = l .
p i ( s ) = ( d + l] (i+ l)'/2 t°d-~ i \ i + 1 / -i !-2"-~ ,o--.
"
Po(s) = (d + 1)
[ 1 [ e ,~d/2
Now we also prove a respective estimate in the intermediate cases.
Let
1 K = 1 + 24-~e"
Finite Sphere Packing and Sphere Covering 33
Lemma 2. Let s ~ R, s >- O, to = (es2)l/4[(47rd) I/4+ (es2)l/4]
-I. Then:
(a) Pi(s) = O(d[1 + (es2/47rd)l/4] TM) for 1 <-- i <-- d - 1.
(b) Pi( s ) = O( d[ 2( s2 / d)~/2] d ) for Kd <~ i <- d -
1.
Proof. Setting A = i /d , by Stirling's fo rmula we have
d3d/2ei/2 P~(s)=O(i3/2~i ) i2,(d_i)3(d-O/22,Tri/2s~ )
= 0 "~I/2A3/2(I_A ) 2~r , /2 da/2A2~(1 ). - - A 3 ( I - A ) / 2
"
The first factor is maximal for A = 1/d, thus
1
Now, for 0 < t < 1, let
t es 2 y ( t ) = ~ In 4 - - ~ - 2t In t - 2(1 - t) In(1 - t).
Then the second factor is b o u n d e d by e v~x)d. Fur thermore ,
we have
y ' ( t ) =½In - 2 I n t + 2 l n ( 1 - t),
y"(0 -- -2 \ i i - ~ / < 0.
Thus y is a concave funct ional which is max imal for to. So, with
some calculat ion, we have
y ( t )~y ( to )=21nr l [ [ es2 \i/4-] J
Putting things together, we obta in the first asserted equali ty
(a). Now, for to <- t < 1, y ( t ) is decreasing. On the
other hand, to = o(K) and, with some fur ther calculat ion,
2 r / e s 2 \ I/4 "1 in ~ - J Y('C)=l+2,/-;eL t 4 ¢ r d ) 2 v / - ~
e l n ( 2 v ~ e ) +21n( l+2x/ - -~e)
-< In 2 + ~ In s~, 2 d
thus
for Kd _< i -< d - 1, which also proves (b). [ ]
34 G. Fejes T6th, P. Gritzmann, and J. M. Wills
Using L e m m a 2 and the trivial b o u n d a~ ~< 1 we obtain an
u p p e r bound for V(sTa+ Ba). But we can do even bet ter i f we
take Hadwiger ' s [13] est imate
oq < 2 - In 2,¢r~ i
for 1 -< i < -- Kd. As Hadwige r [13] fur ther shows
Pi(s)cti < - 1 ( d + v / - d ) 1 tad-i d a./27 i ..... sio(dJg).
(2d) i/2 d / tad
Using Stirl ing's fo rmula again we obtain the fol lowing est imate
for P~(s)ai,
l<_i<_Kd, s<_2q~.
Lemma 3. Let s ~ R, 0 <- s <- 2~-~, 1 <- i <- Kd.
Then
P~( s )a~ = o( d47).
Proof. Setting A = i / d we have
1 [ 1 ~d+J-a'[ d ~(d-i)/2SiO(d#-j) P ' ( s ) a ' - 2 ' e ' / 2 z r
' / 2 t l + ' ~ ) t - d ~ )
1 [ I \ ( t -x) /2 -la = ( 4 7 r ~ ) ~ / 2 [ ~ [ ' ~ ) sXj
o(d'/'d).
s z 1 - t i n ( l _ t). Y(t)=21n4~re 2
Now, for 0 - t - K, let
Then, o f course,
Pi( s )vq = eV(~ )d o( d4-ff ).
Fur thermore , since y ' ( t )= ½(ln(s2/4~r)+ I n ( 1 - t)) the
funct ional y is decreasing, thus
y(t)<-y(O) = 0 ,
Finite Sphere Packing and Sphere Covering 35
% Proof of Theorem 3
Proof of Part (a). Let T= cony(A) and let 7". denote those parts of
T+ B ~ which are contained in the sum of an n-dimensional face of T
and the associated (d-n)-dimensional cone of outer normals.
Furthermore, let K . = T . ~ L_j{a + B "~ la c A}. Then, by means
of Section 3, we have
V(T+ Ba)= ~ VtT.)
= ,o .+ V(K. ) wd "¢0"> ~o~ + Y . VIK.) n - I O ' n ~ d O ) d n
~ I
= V(S,.,+B"),
Proof of Part (b).
Using the notation of Section 6 we have
V(sT+B ~) ! (1+ ~ e, ts)a,). I(~,~,)= (d+l)co,t = d + l ,~1
Now let O~ s-~ 24"~. Then, essentially by Lemmas 2(b) and 3, we
obtain
~r ( ~ . ) = o(d'd') .
Furthermore, if O~ s,~ vTd/13, then, by [.emma 2(a),
P,(s) O ( d [ I (13 x41 r ) ' / ' ] TM) = + : e = o ( 2 ~ ~,)
for i ~ i ~ d - ! and also
thus
[ ]
D
8. A Lower Bound for the Volume of Parallel Bodies of the Regular
Simplex
For the proo f o f Theorem 4 we need a lower est imate for the vo
lume o f parallel bodies of the regular simplex. Let s, r ~ R, s, r
~ 0. As in Section 6 we have
' ( , + , ) , . F • v(sr"+,8")= o\i+l
36 G. Fejes T6th, P. Gritzmann, and J. M. Wills
As we have pointed out in Section 6, Hadwiger [ 13] gave upper
estimates of the external angles ai which, in particular, for i =
1, d >- 3, read as follows:
oq < 2 2v/2--~ d 1 ~ _ i x/ln(d - 1) - In 2x/~ = O ( - - ~ )
.
We now deduce a lower bound of type O((ln d)~/4/d2).
Lemma 4. Let d >- 3. Then we have
1 (In d) 1/4
2 ~ d ( d + l ~ < a ' '
Proof. Using results of Ruben [21] and Hadwiger [13] we have
oq =x/2 [ l f ( t ) t d - t dt, 30
where
f ( t ) = e -h2('f-~(t-l/2)), 0 -< t -< 1
(with the usual convention e - w = 0) a n d h(z) is defined
via
f~ (z) = Z. e-e 2 dt
Since
f " ( t) = -21r e h~<'/~('-)/2~) < O,
thus f is a concave functional of t. Furthermore, f is symmetrical
with respect to ½, i.e.,
f ( t ) = f ( 1 - t ) and f (O)=O, f ( ½ ) = l .
So, for every point to, O< to<½, we have
g~(t)<--f( t) for 0--< t-< 1,
where
and
to gt°(t) = . 2 ( f ( to )~ 1)
[. 1 -2 to ( ½ - t ) + l
g~(t) = g,o(1 - t) .
for O-< t-< to,
Finite Sphere Packing and Sphere Covering 37
In order to deduce a lower bound for f ( to) we have to give an
upper estimate for h(x/-~r(½-to)), which, in view of the
inequality
x / ~ e-Z2 f o 2 2z -~ e -'2 dt,
means finding solutions Xo of
Setting
2 2Xo
1 to - 2x/-~d' Xo = [In d -¼ In(In d)] '12
we have
x 2 1 d)l/4<l In(In e- o= ~ (In [In d -~ d)] 1/2 =
2x/-~toXo
and thus
h2(vC-~(½ - to)) <- In d --~ In(In d).
So, with some calculation we obtain
a~= x/2 f¢ f ( t ) t a-t dt>_ x/2 f(t°-----~) f~ ( 1 - t ) t d-~
dt to 1 - to
1 I = ~ f ( t o ) [ l _ ( l _ t o ) a ] _ _ d _ _ ~ [ l _ ( l _ t o
) a + , ] to
1 1 3 In d > 2 2 x / 2 ~ [ 1 - ( 1 + ~ - - ~ ) ( 1 - ~ - ~ ) ] d
( d + l )
1 (In d) l/4
2x/5 d (d + 1)"
Let us point out that this estimate can easily be improved for
small dimensions, but here we are mainly interested in its
asymptotical behavior• []
Using Lemma 4 we at once obtain a suitable lower bound for V ( s T
a + Bd).
Corollary. Let d >- 3, r, s ~ R, r, s >- O• Then
a 1 roar +-7-'~. (in d)l/4tOd_lrd-ls <-- V ( s T d + rBa).
4V~
38 G. Fejes T6th, P. Gritzmann, and J. M. Wills
9. Proof of Theorem 4
The proof is an application of the previous section and the
following covering result.
Lemma 5. Let ro = (I- I/d) ~/2. Then
soTd +ro Bac U a+Bd. a e - A 0
Proof. As is well known (and easily calculated) the circumsphere
radius of the n-dimensional regular simplex of side 1 is (n/[2(n +
1)]) I/2. Thus, since So< 1, in particular we have
and trivially
So Td C U (a+Bd) a ~ A o
U (a+roBa) c U (a+Ba) • a ~ A o a E A o
Now let 1 - n <- d - 1 and let f be an n-dimensional face of So
Td. Furthermore, let p be an outer normal of So Ta (taken at 0)
with length r o. Then it is sufficient to prove the following
inclusion;
p + f c U ( a + B d ) • a ~ AoC~f
Let a ~ Ao n f and E = af t(p + f ) . Then (a + B d) n E is an
n-dimensional ball of radius ( 1 - r2) l/2. Since p + f is itself
an n-dimensional simplex of side So the assertion follows
from
[ n rob,/ "
By Lemma 5 and the corollary to Lemma 4 we have
V(soTd+ro Bd) 1 1 ( l nd ) I/4 > ro d ~ _ _ O)d"- 1 d-I ( d + l
) t o a d + l 4x/5 d + l rod ro So.
O.)d_ I ) ~ O)d'
y( Ao) < 4 - -
d+1
(Ind) I/4'
10. Proof of Theorem 5
Let Q be the octahedron
Q = 2 conv{(u, u, 0), (u, -u , 0), ( -u , u, 0 ) ( -u , -u , 0),
(0, 0, u), (0, 0, -u )}
and let s e N. Let G(s) denote the number of lattice points of q3
contained in sQ. Then
.3-1 s - I
G ( s ) = ( 2 s + l ) 2 + 2 ~, (2 i+1)2+2 Y~ (2 i+2) 2 i=O
i=O
= ~ s3+8s2 +~ s+ l.
After some easy calculations we obtain, for the intrinsic volumes
of Q,
Vo(Q) = 1,
8 VI(Q) = 4 x ~ x ~+ 8 x 4 x ~, = ~ (2+V~) ,
16xx/2 1 64x/2 V2(Q) = 8 - -
3 2 - 3 "
512 ~ ( Q ) - 9x/3"
and so
8 V(sQ+Ba)=__f+Ir_~(2+x/~)s+2x 6 2 512s3
V ( s Q + B d ) _ v ( s c ~ , ) + B d. 3 2 [ 16 \ 3 /8v/2 \ 2 ) = -
~ ~ - ~ - ¢r)s + 16~--~--- ¢r)s
4¢r +-~- (4 + 2 , / 3 - 7)s.
Therefore we have
G ( s ) > 0 for s<-15, G ( s ) < 0 for s->16
with
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Received June 19, 1986.