finite sphere packing and sphere covering*
TRANSCRIPT
19_1.tifDiscrete Comput Geom 4:19-40 (1989) C Diserete &
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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
References
1. U. Betke and P. Gritzmann, 0ber L. Fejes T6ths Wurstvermutung in kleinen Dimensionen, Acta Math. Hungar. 43 (1984), 299-307.
40 G. Fejes T6th, P. Gritzmann, and J. M. Wills
2. U. Betke, P. Gritzmann, and J. M. Wills, Slices ofL. Fejes T6th's sausage conjecture, Mathematika 29 (1982), 194-201.
3. H. F. Blichfeldt, The minimum value of quadratic forms and the closest packing of spheres, Math. Ann. 101 (1929), 605-608.
4. H. S. M. Coxeter, L. Few, and C. A. Rogers, Covering space with equal spheres, Mathematika 6 (1959), 147-157.
5. L. Fejes T6th, I]ber die dichteste Kreislagerung und diinnste Kreisiiberdeckung, Comment. Math. Heir. 23 (1949), 342-349.
6. L. Fejes T6th, Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, 1953, 1972.
7. L. Fejes T6th, Research problem 13, Period. Math. Hungar. 6 (1975), 197-199. 8. G. Fejes T6th, New results in the theory of packing and covering, in Convexity and Its Applications
(P. M. Gruber and J. M. Wills, eds.), 318-359, Birkh~iuser, Basel, 1983. 9. P. Gritzmann and J. M. Wills, On two finite covering problems of Bambah, Rogers, Woods, and
Zassenhaus, Monatsh. Math. 99 (1985), 279-296. 10. P. Gritzmann and J. M. Wills, Finite packing and covering, Studia Sci. Math. Hungar. 21 (1986),
151-164. 11. H. Groemer, Uber die Einlagerung von Kreisen in einen konvexen Bereich, Math. Z. 73 (1960),
285-294. 12. H. Hadwiger, Vorlesungen iiberlnhalt, Oberfliicheundlsoperimetrie, Springer-Verlag, Berlin, 1957. 13. H. Hadwiger, Gitterpunktanzahl im Simplex und Wills'sche Vermutung, Math. Ann. 239 (1979),
271-288. 14. G. A. Kabatjanski and V. I. Levenstein, Bounds for packings on the sphere and in space, Problemy
Peredachi lnformaticii 14 (1978), 3-25 (Russian). (English transl.; Problems Inform. Transmission 14 (1978), 1-17.)
15. J. Leech, Some sphere packings in higher space, Canad. J. Math. 16 (1964), 657-682. 16. P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes, Math. Proc.
Cambridge Philos. Soc. 78 (1975), 247-261. 17. A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch
a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), 210-214. 18. R. A. Rankin, The closest packing of spherical caps in n dimensions, Proc. Glasgow Math. Assoc.
2 (1955), 139-144. 19. C. A. Rogers, The packing of equal spheres, Proc. London Math. Soc. (3), 8 (1958), 609-620. 20. C. A. Rogers, Packing and Covering, Cambridge University Press, Cambridge, 1964. 21. H. Ruben, On the geometrical moments of skew-regular simplices in hyperspherical space; with
some applications in geometry and mathematical statistics, Acta Math. 103 (1960), 1-23. 22. G. Wegner, 1]ber endliche Kreispackungen in der Ebene, Studia Sci. Math. Hungar., in press. 23. J. M. Wills, Research problems 30, 33, and 35, Period. Math. Hungar. 13 (1982), 75-76, 14 (1983),
189-191,312-314. 24. J. M. Wills, On the density of finite packings, Acta Math. Hungar. 46 (1985), 205-210. 25. J. M. Wills, Research problem 41: space conjecture for finite packing and covering, Period. Math.
Hungar. 18 (1987), 251-252.
Received June 19, 1986.
, 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
References
1. U. Betke and P. Gritzmann, 0ber L. Fejes T6ths Wurstvermutung in kleinen Dimensionen, Acta Math. Hungar. 43 (1984), 299-307.
40 G. Fejes T6th, P. Gritzmann, and J. M. Wills
2. U. Betke, P. Gritzmann, and J. M. Wills, Slices ofL. Fejes T6th's sausage conjecture, Mathematika 29 (1982), 194-201.
3. H. F. Blichfeldt, The minimum value of quadratic forms and the closest packing of spheres, Math. Ann. 101 (1929), 605-608.
4. H. S. M. Coxeter, L. Few, and C. A. Rogers, Covering space with equal spheres, Mathematika 6 (1959), 147-157.
5. L. Fejes T6th, I]ber die dichteste Kreislagerung und diinnste Kreisiiberdeckung, Comment. Math. Heir. 23 (1949), 342-349.
6. L. Fejes T6th, Lagerungen in der Ebene, auf der Kugel und im Raum, Springer-Verlag, Berlin, 1953, 1972.
7. L. Fejes T6th, Research problem 13, Period. Math. Hungar. 6 (1975), 197-199. 8. G. Fejes T6th, New results in the theory of packing and covering, in Convexity and Its Applications
(P. M. Gruber and J. M. Wills, eds.), 318-359, Birkh~iuser, Basel, 1983. 9. P. Gritzmann and J. M. Wills, On two finite covering problems of Bambah, Rogers, Woods, and
Zassenhaus, Monatsh. Math. 99 (1985), 279-296. 10. P. Gritzmann and J. M. Wills, Finite packing and covering, Studia Sci. Math. Hungar. 21 (1986),
151-164. 11. H. Groemer, Uber die Einlagerung von Kreisen in einen konvexen Bereich, Math. Z. 73 (1960),
285-294. 12. H. Hadwiger, Vorlesungen iiberlnhalt, Oberfliicheundlsoperimetrie, Springer-Verlag, Berlin, 1957. 13. H. Hadwiger, Gitterpunktanzahl im Simplex und Wills'sche Vermutung, Math. Ann. 239 (1979),
271-288. 14. G. A. Kabatjanski and V. I. Levenstein, Bounds for packings on the sphere and in space, Problemy
Peredachi lnformaticii 14 (1978), 3-25 (Russian). (English transl.; Problems Inform. Transmission 14 (1978), 1-17.)
15. J. Leech, Some sphere packings in higher space, Canad. J. Math. 16 (1964), 657-682. 16. P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes, Math. Proc.
Cambridge Philos. Soc. 78 (1975), 247-261. 17. A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch
a unit sphere in n dimensions, J. Combin. Theory Ser. A 26 (1979), 210-214. 18. R. A. Rankin, The closest packing of spherical caps in n dimensions, Proc. Glasgow Math. Assoc.
2 (1955), 139-144. 19. C. A. Rogers, The packing of equal spheres, Proc. London Math. Soc. (3), 8 (1958), 609-620. 20. C. A. Rogers, Packing and Covering, Cambridge University Press, Cambridge, 1964. 21. H. Ruben, On the geometrical moments of skew-regular simplices in hyperspherical space; with
some applications in geometry and mathematical statistics, Acta Math. 103 (1960), 1-23. 22. G. Wegner, 1]ber endliche Kreispackungen in der Ebene, Studia Sci. Math. Hungar., in press. 23. J. M. Wills, Research problems 30, 33, and 35, Period. Math. Hungar. 13 (1982), 75-76, 14 (1983),
189-191,312-314. 24. J. M. Wills, On the density of finite packings, Acta Math. Hungar. 46 (1985), 205-210. 25. J. M. Wills, Research problem 41: space conjecture for finite packing and covering, Period. Math.
Hungar. 18 (1987), 251-252.
Received June 19, 1986.