Sphere Packings

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Sphere Packings

Chuanming Zong

Springer

To Peter M. Gruber and David G. Larman

This page intentionally left blank

Preface

Sphere packings is one of the most fascinating and challenging subjects in mathematics. Almost four centuries ago, Kepler studied the densities of sphere packings and made his famous conjecture. Several decades later, Gregory and Newton discussed the kissing numbers of spheres and proposed the Gregory-Newton problem. Since then, these problems and related ones have attracted the attention of many prominent mathematicians such as Blichfeldt, Dirichlet, Gauss, Hermite, Korkin, Lagrange, Minkowski, Thue, Voronoi, Watson, and Zolotarev, as well as many active today. As work on the classical sphere packing problems has progressed many exciting results have been obtained, ingenious methods have been created, related challenging problems have been proposed and investigated, and surprising connections with other subjects have been found. Thus, though some of its original problems are still open, sphere packings has developed into an important discipline. There are several books dealing with various aspects of sphere packings. For example, Conway and Sloane [1], Erd¨ os, Gruber, and Hammer [1], L. Fejes T´ oth [9], Gruber and Lekkerkerker [1], Leppmeier [1], Martinet [1], Pach and Agarwal [1], Rogers [14], Siegel [2], and Zong [4]. However, none of them gives a full account of this fascinating subject, especially its local aspects, discrete aspects, and proof methods. The purpose of this book is try to do this job. It deals not only with the classical sphere packing problems, but also the contemporary ones such as blocking light rays, the holes in sphere packings, and ﬁnite sphere packings. Not only are the main results of the subject presented, but also, its creative methods from areas such as geometry, number theory, and linear programming are described.

viii

Preface

In addition, the book contains short biographies of several masters of this discipline and also many open problems. I am very much indebted to Professors P.M. Gruber, T.C. Hales, M. Henk, E. Hlawka, D.G. Larman, C.A. Rogers, J.M. Wills, G.M. Ziegler, and the reviewers for their helpful information, suggestions, and comments to the manuscript, and to J. Talbot for his wonderful editing work. Nevertheless, I assume sole responsibility for any remaining mistakes. The staﬀ at Springer-Verlag in New York have been courteous, competent, and helpful, especially M. Cottone, K. Fletcher, D. Kramer, and Dr. I. Lindemann. Also, I am very grateful to my wife, Qiaoming, for her understanding and support. This work is supported by The Royal Society, the Alexander von Humboldt Foundation, and the National Scientiﬁc Foundation of China.

Berlin, 1999

C. Zong

Basic Notation

En x o x, y x, y x d(X) conv{X} K int(K) rint(K) bd(K) v(K) s(K) C x, yC Sn ωn x, yg In Z

Euclidean n-dimensional space. A point (or a vector) of E n with coordinates (x1 , x2 , . . . , xn ). The origin of E n . The inner product of two vectors x and y. The Euclidean distance between two points x and y. The Euclidean norm of x. The diameter of a set X. The convex hull of X. An n-dimensional convex body. The interior of K. The relative interior of K in the considered space. The boundary of K. The volume of K. The surface area of K. An n-dimensional centrally symmetric convex body centered at o. The Minkowski distance between x and y with respect to C. The n-dimensional unit sphere centered at o. The volume of the n-dimensional unit sphere. The geodesic distance between two points x and y in bd(Sn ). The n-dimensional unit cube {x : |xi | ≤ 12 }. The set of all integers.

x Zn Λ det(Λ) Q(x) γn ℵ x, yH D(x) k(Sn ) k ∗ (Sn ) b(Sn ) δ(Sn ) δ ∗ (Sn ) θ(Sn ) θ∗ (Sn ) r(Sn ) r∗ (Sn ) ρ∗ (Sn ) h(Sn ) h∗ (Sn ) (Sn )

Basic Notation The n-dimensional integer lattice {z : zi ∈ Z}. An n-dimensional lattice. The determinant of Λ. A positive deﬁnite quadratic form. The Hermite constant. A code. The Hamming distance between x and y. The Dirichlet-Voronoi cell at x. The kissing number of Sn . The lattice kissing number of Sn . The blocking number of Sn . The maximum packing density of Sn . The maximum lattice packing density of Sn . The minimum covering density of Sn . The minimum lattice covering density of Sn . The maximum radius of a sphere that can be embedded into every packing of Sn . The maximum radius of a sphere that can be embedded into every lattice packing of Sn . The maximum radius of the spherical base of an inﬁnite cylinder that can be embedded into every lattice packing of Sn . The Hornich number of Sn . The lattice Hornich number of Sn . The L. Fejes T´ oth number of Sn .

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.

2.

vii ix

The Gregory–Newton Problem and Kepler’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1. 1.2. 1.3. 1.4. 1.5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packings of Circular Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gregory-Newton Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . Kepler’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 7 10 13 18

Positive Deﬁnite Quadratic Forms and Lattice Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Lagrange-Seeber-Minkowski Reduction and a Theorem of Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Mordell’s Inequality on Hermite’s Constants and a Theorem of Korkin and Zolotarev . . . . . . . . . . . . . . . . . . 2.4. Perfect Forms, Voronoi’s Method, and a Theorem of Korkin and Zolotarev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. The Korkin-Zolotarev Reduction and Theorems of Blichfeldt, Barnes, and Vet˘cinkin . . . . . . . . . . . . . . . . . . . . . . 2.6. Perfect Forms, the Lattice Kissing Numbers of Spheres, and Watson’s Theorem . . . . . . . . . . . . . . . . . . . . . . .

23 25 31 33 36 41

xii

Contents 2.7. Three Mathematical Geniuses: Zolotarev, Minkowski, and Voronoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.

4.

Lower Bounds for the Packing Densities of Spheres . . . . 3.1. The Minkowski-Hlawka Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Siegel’s Mean Value Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Sphere Coverings and the Coxeter-Few-Rogers Lower Bound for δ(Sn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Edmund Hlawka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lower Bounds for the Blocking Numbers and the Kissing Numbers of Spheres . . . . . . . . . . . . . . . . . . . . . 4.1. The Blocking Numbers of S3 and S4 . . . . . . . . . . . . . . . . . . . . . 4.2. The Shannon-Wyner Lower Bound for Both b(Sn ) and k(Sn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. A Theorem of Swinnerton-Dyer . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. A Lower Bound for the Translative Kissing Numbers of Superspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 47 47 51 55 62

65 65 71 72 74

5.

Sphere Packings Constructed from Codes . . . . . . . . . . . . . . . 5.1. Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Construction A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Construction B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Construction C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Some General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 82 84 85 89

6.

Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres I . . . . . . . . . . . . . . . . . . . 91 6.1. Blichfeldt’s Upper Bound for the Packing Densities of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2. Rankin’s Upper Bound for the Kissing Numbers of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.3. An Upper Bound for the Packing Densities of Superspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4. Hans Frederik Blichfeldt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.

Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres II . . . . . . . . . . . . . . . . . . 7.1. Rogers’ Upper Bound for the Packing Densities of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Schl¨ aﬂi’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. The Coxeter-B¨ or¨ oczky Upper Bound for the Kissing Numbers of Spheres . . . . . . . . . . . . . . . . . . . . . . 7.4. Claude Ambrose Rogers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 107 111 122

Contents 8.

9.

Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres III . . . . . . . . . . . . . . . . . 8.1. Jacobi Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Delsarte’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. The Kabatjanski-Leven˘stein Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

125 125 127

132

The Kissing Numbers of Spheres in Eight and Twenty–Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Some Special Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Two Theorems of Leven˘stein, Odlyzko, and Sloane . . . . . . . 9.3. Two Principles of Linear Programming . . . . . . . . . . . . . . . . . . . 9.4. Two Theorems of Bannai and Sloane . . . . . . . . . . . . . . . . . . . . .

139 139 141 142 143

10. Multiple Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. A Basic Theorem of Asymptotic Type . . . . . . . . . . . . . . . . . . . 10.3. A Theorem of Few and Kanagasahapathy . . . . . . . . . . . . . . . 10.4. Remarks on Multiple Circle Packings . . . . . . . . . . . . . . . . . . . .

153 153 154 157 162

11. Holes in Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Spherical Holes in Sphere Packings . . . . . . . . . . . . . . . . . . . . . . 11.2. Spherical Holes in Lattice Sphere Packings . . . . . . . . . . . . . . 11.3. Cylindrical Holes in Lattice Sphere Packings . . . . . . . . . . . .

165 165 176 178

12. Problems of Blocking Light Rays . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. Hornich’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3. L. Fejes T´ oth’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4. L´ aszl´ o Fejes T´ oth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 185 189 198

13. Finite Sphere Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. The Spherical Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3. The Sausage Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4. The Sausage Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 200 204 214

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

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1. The Gregory–Newton Problem and Kepler’s Conjecture

1.1. Introduction Let K denote a convex body in n-dimensional Euclidean space En . In other words, K is a compact subset of E n with nonempty interior such that λx + (1 − λ)y ∈ K, whenever both x and y belong to K and 0 < λ < 1. We call K a centrally symmetric convex body if there is a point p such that x ∈ K if and only if 2p − x ∈ K. For example, both the n-dimensional unit sphere Sn =

n x = (x1 , x2 , . . . , xn ) : (xi )2 ≤ 1 i=1

and the n-dimensional unit cube 1 In = x = (x1 , x2 , . . . , xn ) : max |xi | ≤ 1≤i≤n 2 are centrally symmetric convex bodies. As usual, the interior, boundary, volume, surface area, and diameter of K are denoted by int(K), bd(K), v(K), s(K), and d(K), respectively. Denoting the volume of Sn by ωn , it is well-known that π n/2 ωn = , Γ(n/2 + 1) and therefore s(Sn ) = nωn =

nπ n/2 , Γ(n/2 + 1)

2

1. The Gregory-Newton Problem, Kepler’s Conjecture

where Γ(x) is the gamma function. If ai = (ai1 , ai2 , . . . , ain ), i = 1, 2, . . . , n, are n linearly independent vectors in E n , then the set Λ=

n

zi ai : zi ∈ Z ,

i=1

where Z is the set of all integers, is called a lattice, and we call {a1 , a2 , . . . , an } a basis for Λ. As usual, the absolute value of the determinant |aij | is called the determinant of the lattice and is denoted by det(Λ). Geometrically speaking, det(Λ) is the volume of the fundamental parallelepiped P =

n

λi ai : 0 ≤ λi ≤ 1

i=1

of Λ. Let X be a set of discrete points in En . We shall call K +X a translative packing of K if (int(K) + x1 ) ∩ (int(K) + x2 ) = ∅ whenever x1 and x2 are distinct points of X. In particular, we shall call it a lattice packing of K when X is a lattice. Let k(K) and k∗ (K) denote the translative kissing number and the lattice kissing number of K, respectively. In other words, k(K) is the maximum number of nonoverlapping translates K + x that can touch K at its boundary, and k∗ (K) is deﬁned similarly, with the restriction that the translates are members of a lattice packing of K. For every convex body K, it is easy to see that k ∗ (K) ≤ k(K). (1.1) The following result provides a general upper bound for k∗ (K) and k(K). Theorem 1.1 (Minkowski [7] and Hadwiger [1]). Let K be an ndimensional convex body. Then k ∗ (K) ≤ k(K) ≤ 3n − 1, where equality holds for parallelepipeds. Proof. The parallelepiped case is easy to verify. We omit the veriﬁcation here. Let D(A) denote the diﬀerence set of a convex set A. In other words, D(A) = {x − y : x, y ∈ A}.

1.1. Introduction

3

By convexity, it is easy to see that (A + x1 ) ∩ (A + x2 ) = ∅ if and only if

1

2 D(A)

+ x1 ∩ 12 D(A) + x2 = ∅.

Therefore, by considering the two cases A = int(K) and A = K it can be easily deduced that k(K) = k(D(K)). Since D(K) is centrally symmetric, in order to get an upper bound for k(K), we assume that K itself is centrally symmetric and centered at o. If a translate K + x touches K at y ∈ bd(K), then for any point z ∈ K + x, we have 3y ∈ 3K and 3 2 (z

− y) ∈ 3K.

Hence we have z=

1 3

× 3y +

2 3

× 32 (z − y) ∈ 3K,

and therefore (see Figure 1.1) K + x ⊆ 3K.

3K K

K +x

Figure 1.1 Let K + xi , i = 1, 2, . . . , k(K), be nonoverlapping translates that touch K at its boundary. Then

k(K)

i=0

(K + xi ) ⊆ 3K,

4

1. The Gregory-Newton Problem, Kepler’s Conjecture

where x0 = o, and therefore (k(K) + 1)v(K) ≤ v(3K) = 3n v(K), which implies that k(K) ≤ 3n − 1, 2

and hence using (1.1), Theorem 1.1 is proved.

Remark 1.1. In fact, n-dimensional parallelepipeds are the only convex bodies in En such that k ∗ (K) = k(K) = 3n − 1. For a proof see Hadwiger [1] and Groemer [2]. Let l be a positive number and let m(K, l) be the maximum number of translates K + x that can be packed into the cube lIn . We deﬁne δ(K) = lim sup l→∞

m(K, l)v(K) , v(lIn )

the density of the densest translative packings of K in En . Similarly, the density δ∗ (K) of the densest lattice packings of K is deﬁned by restricting the translative vectors to those in a lattice. In this case, it can be deduced that v(K) δ ∗ (K) = sup , Λ det(Λ) where the supremum is over all lattices Λ such that K + Λ is a packing. For every convex body K, it is easy to see that δ ∗ (K) ≤ δ(K) ≤ 1.

(1.2)

The density of a given translative packing K + X is deﬁned by δ(K, X) = lim sup l→∞

card{X ∩ lIn }v(K) . v(lIn )

Then simple analysis yields that δ(K) = sup δ(K, X), X

where the supremum is over all sets X such that K + X is a packing. In fact, as the following theorem shows, the density δ(K) can be deﬁned using any convex body containing the origin, not just the cube. Theorem 1.2 (Hlawka [4]). Let A be an arbitrary convex body that contains the origin o as an interior point, and let m(K, lA) be the maximum number of translates K + x that can be packed into lA. Then, lim

l→∞

m(K, lA)v(K) = δ(K). v(lA)

1.1. Introduction

5

Proof. Let m∗ (K, l) be the maximum number of nonoverlapping translates K + x that intersect lIn and let d be the diameter of K. Since v((l + 2d)In ) − v((l − 2d)In ) = 0, l→∞ v(lIn ) lim

(1.3)

from the deﬁnition of δ(K) it follows that for each > 0 there is a t such that m(K, t )v(K) > δ(K) − v(t In ) and

m∗ (K, t )v(K) < δ(K) + . v(t In )

On the other hand, for any convex body A, there exists an l > 0 such that for each l > l there are two families of nonoverlapping cubes, t l In + xi : i = 1, 2, . . . , g and

t l

such that

In + yi : i = 1, 2, . . . , h , g t

In + xi ⊆ A, l v ∪gi=1 tl In + xi v(A) ≤ , 1−

h t In + yi , A⊆ l i=1 i=1

v ∪hi=1 tl In + yi v(A) ≥ . 1+ Hence, when l > l , one has

and

(1 − )(δ(K) − ) <

m(K, lA)v(K) < (1 + )(δ(K) + ), v(lA)

which implies the assertion of Theorem 1.2.

2

Remark 1.2. Clearly, k(K), k∗ (K), δ(K), and δ∗ (K) are invariant under nonsingular linear transformations. Suppose that X = {x1 , x2 , . . . } is a discrete set of points in E n . For each point xi ∈ X the Dirichlet-Voronoi cell D(xi ) is deﬁned by D(xi ) = {x : x, xi ≤ x, xj for all xj ∈ X} ,

6

1. The Gregory-Newton Problem, Kepler’s Conjecture

where x, y indicates the Euclidean distance between x and y. In other words, D(xi ) = x : x − xi , xj − xi ≤ 12 xj + xi , xj − xi , xj ∈X

where · denotes the inner product. Thus every Dirichlet-Voronoi cell is a convex polytope. The Dirichlet-Voronoi cell has many basic properties. For example, it can be easily deduced from its deﬁnition that En = D(xi ), xi ∈X

and int(D(xj )) ∩ int(D(xk )) = ∅ for every pair of distinct points xj and xk of X. In other words, the cells D(xi ) form a tiling of E n . Thus, if K + X is a packing, D(xi ) can be regarded as a local cell associated to K + xi . In particular, when K is a sphere centered at o, then K + xi ⊂ D(xi ) for every point xi ∈ X. By the deﬁnition of δ(K), for any > 0, there exist a packing K + X and a positive number c such that d(D(xi )) < c for every point xi ∈ X, and card{X ∩ lIn }v(K) ≥ δ(K) − . l→∞ v(lIn ) lim

Then, by (1.3), one has δ(K) ≤ for suﬃciently large l, where λ(X, l) =

v(K) + 2 λ(X, l)

(1.4)

v(D(xi )) . card{X ∩ lIn }

xi ∈X∩lIn

In particular, since all the Dirichlet-Voronoi cells are congruent if X is a lattice, one has v(K) δ ∗ (K) = sup , (1.5) Λ v(D(o)) where the supremum is over all lattices Λ such that K + Λ is a packing. Clearly, (1.4) and (1.5) provide reasonable ways to approximate δ(K) and δ ∗ (K) using local computation.

1.2. Packings of Circular Disks

7

1.2. Packings of Circular Disks Lemma 1.1. Let P be a polygon with p edges containing the unit disk S2 . Then, π v(P ) ≥ p tan , p where equality holds if and only if P is regular and its edges are tangent to bd(S2 ). Proof. Without loss of generality, we may assume that the p edges L1 , L2 , . . . , Lp of P are tangent to bd(S2 ) at t1 , t2 , . . . , tp in a circular order (see Figure 1.2). Letting θi be the angle between ti and ti+1 , where tp+1 = t1 , it follows that p θi = 2π i=1

and v(P ) =

p

tan

i=1

θi . 2

t3

S2

θ2 θ1

t2

t1 Figure 1.2 Since the function f (x) = tan x is strictly concave on [0, π/2), by Jensen’s inequality it follows that

p θi π v(P ) ≥ p tan i=1 = p tan , 2p p where equality holds if and only if P is a regular polygon. Lemma 1.1 is proved. 2 √ Let Λ2 be the two-dimensional lattice with basis {(2, 0), (1, 3)}. Then S2 + Λ2 is a lattice packing of S2 . From this example it follows that k ∗ (S2 ) ≥ 6

(1.6)

8

1. The Gregory-Newton Problem, Kepler’s Conjecture

and

v(S2 ) π =√ . (1.7) det(Λ2 ) 12 On the other hand, if S2 +x1 and S2 +x2 are two nonoverlapping circular disks that touch S2 at its boundary, then the angle between x1 and x2 is at least π/3. Therefore, 2π k(S2 ) ≤ = 6. (1.8) π/3 δ ∗ (S2 ) ≥

From (1.1), (1.6), and (1.8) it follows that k ∗ (S2 ) = k(S2 ) = 6. The following result determines the exact values of δ∗ (S2 ) and δ(S2 ). Theorem 1.3 (Lagrange [1] and Thue [2]). π δ ∗ (S2 ) = δ(S2 ) = √ . 12 Proof. Let X be a set such that S2 + X is a packing and d(D(x)) ≤ c for every point x ∈ X, where c is a suitable positive number. By (1.4) v(S2 ) + 2 λ(X, l)

δ(S2 ) ≤

(1.9)

for suﬃciently large l, where

λ(X, l) =

v(D(x)) . card{X ∩ lI2 } x∈X∩lI2

(1.10)

For convenience, we write X ∗ = X ∩ (l + 2c)I2 and deﬁne D∗ (x) = {y : y, x ≤ y, w for all w ∈ X ∗ } for each x ∈ X ∗ . It is easy to see that D∗ (x) = D(x),

x ∈ X ∩ lI2 ,

(1.11)

and the D∗ (x) form a tiling of E2 . Let e, f , and v be the number of edges, polygons, and vertices of this tiling. By Euler’s formula, f + v = e + 2.

(1.12)

1.2. Packings of Circular Disks

9

Let V be the set of vertices of this tiling, and let p(x) and q(v) denote the number of edges of D∗ (x) and the number of edges into v ∈ V , respectively. Since every vertex belongs to at least three edges and every edge joins two vertices, we have 3v ≤ q(v) = 2e. (1.13) v∈V

Then from (1.12) and (1.13) it follows that e + 6 ≤ 3f. Writing

(1.14)

p(X, l) =

x∈X∩lI2 p(x) , card{X ∩ lI2 }

(1.3), (1.11), and (1.14) imply that p(X, l) ≤

2e 12 + ≤6− + f f

for suﬃciently large l, and hence lim sup p(X, l) ≤ 6.

(1.15)

l→∞

By (1.10) and Lemma 1.1 it follows that λ(X, l) ≥

π

x∈X∩lI2

p(x) π

π tan p(x)

card{X ∩ lI2 }

.

Since the function

x π tan π x is concave when x ≥ 3, by Jensen’s inequality and (1.15) it follows that f (x) =

λ(X, l) ≥ π Hence by (1.9) one has

√ p(X, l) π tan ≥ 12. π p(X, l) π δ(S2 ) ≤ √ . 12

Then Theorem 1.3 follows from (1.2), (1.7), and (1.16).

(1.16) 2

Remark 1.3. From the proof of Theorem 1.3 it follows that the lattice with which π δ ∗ (S2 ) = √ 12 can be realized is unique with respect to rotation and reﬂection.

10

1. The Gregory-Newton Problem, Kepler’s Conjecture

1.3. The Gregory-Newton Problem √ √ √ Write a1 = (2, 0, 0), a2 = (1, 3, 0), and a3 = (1, 3/3, 2 6/3), and let Λ3 be the lattice generated by them. Usually, Λ3 is known as the face-centered cubic lattice. It is easy to see that S3 + Λ3 is a packing of S3 , in which every sphere touches 12 others. This observation implies k ∗ (S3 ) ≥ 12.

(1.17)

In 1694, during a famous conversation, D. Gregory and I. Newton discussed the following problem. The Gregory-Newton Problem. Can a sphere touch 13 spheres of the same size? Newton thought “no, the maximum number is 12,” while Gregory believed the answer to be “yes.” In the literature this problem is sometimes referred to as the thirteen spheres problem. In S3 + Λ3 , locally speaking, the kissing conﬁguration is stable. In other words, none of the twelve spheres that touch S3 can be moved around (see Figure 1.3). However, if twelve unit spheres are placed at positions corresponding to the vertices of a regular icosahedron concentric with the central one, the twelve outer spheres do not touch each other and may all be moved around freely (see Figure 1.4).

Figure 1.3

Figure 1.4

Let S3 + xi , i = 1, 2, . . . , k(S3 ), be nonoverlapping spheres that touch S3 at its boundary, and deﬁne √ Ωi = x ∈ bd(S3 ) : x, xi ≥ 3 .

1.3. The Gregory-Newton Problem

11

The k(S3 ) congruent caps Ωi form a cap packing on the surface of S3 , and therefore, comparing the area of bd(S3 ) with the area of Ωi , it follows that k(S3 ) ≤

s(S3 ) = 14.9282 . . . , s(Ω1 )

which together with (1.17) implies that 12 ≤ k(S3 ) ≤ 14. However, this is not enough to solve the Gregory-Newton problem. These facts show the inherent diﬃculty of this fascinating problem. Theorem 1.4 (Hoppe [1], Sch¨ utte and van der Waerden [1], and Leech [1]). k ∗ (S3 ) = k(S3 ) = 12. Proof. Let X = {x1 , x2 , . . . , xv } be a subset of bd(S3 ) such that S3 + {o, 2x1 , 2x2 , . . . , 2xv } is a kissing conﬁguration, and let xi , xj g be the geodesic distance between xi and xj . Clearly, we have xi , xj g ≥ π/3 whenever xi and xj are distinct points of X. Joining xi and xj by a geodesic arc if and only if xi , xj g < arccos 17 , we obtain a network on the sphere. No two arcs of this network cross, since any four points of the network form a spherical quadrilateral with side length at least π/3 whose diagonals cannot both be less than π/2. Therefore, bd(S3 ) is divided into polygons whose vertices are points of X. Clearly, we may assume that every point of X is connected to at least two others by arcs. An easy computation yields that every angle of these polygons is larger than π/3. Thus at most ﬁve arcs of the network meet in any point of X. Let Pn be a polygon of this network with n sides. By routine computation one can verify the following statements: s(P3 ) ≥ 0.5512 . . . ,

(1.18)

where equality holds if P3 is an equilateral triangle with side length π/3. s(P4 ) ≥ 1.3338 . . . ,

(1.19)

where equality holds if P4 is an equilateral quadrilateral with side length π/3, and one of its diagonals is of length arccos 17 . s(P5 ) ≥ 2.2261 . . . ,

(1.20)

12

1. The Gregory-Newton Problem, Kepler’s Conjecture

where equality holds if P5 is an equilateral pentagon with side length π/3 having two coterminous diagonals of length arccos 17 . Also, when n ≥ 4, it is clear that the excess area of an n-gon over the minimum area of n − 2 triangles increases with n, since if on a side of any n-gon a triangle of minimum area is abutted, the resulting (n + 1)-gon has to be deformed so that the original common side becomes a diagonal of length at least arccos 17 ; consequently, the area is increased. By (1.12), Euler’s formula, one has 2v − 4 = 2e − 2f = 3f3 + 4f4 + 5f5 + · · · − 2(f3 + f4 + f5 + · · ·) = f3 + 2f4 + 3f5 + · · · , where e is the number of the arcs, f is the number of the polygons, and fi is the number of i-gons in the network. Comparing the surface area of S3 and the area of the network, by (1.18), (1.19), and (1.20) it follows that 4π ≥ 0.5512f3 + 1.3338f4 + 2.2261f5 + · · · = 0.5512(f3 + 2f4 + 3f5 + · · ·) + 0.2314f4 + 0.5725(f5 + · · ·) = 0.5512(2v − 4) + 0.2314f 4 + 0.5725(f5 + · · ·). Thus we have 2v − 4 ≤

4π = 22.79, 0.5512

and therefore v ≤ 13. Suppose now that v = 13. Then 2v − 4 = 22, and we have 4π ≥ 0.5512 × 22 + 0.2314f 4 + 0.5725(f5 + · · ·), 0.44 ≥ 0.2314f 4 + 0.5725(f5 + · · ·), and therefore f4 = 0 or 1 and f5 = f6 = · · · = 0. In other words, the network has to divide bd(S3 ) into triangles except possibly for one quadrilateral. Now, we deal with two cases. Case 1. f4 = 0. Then we have f3 = 2e/3, 13 + 2e/3 = e + 2, and therefore e = 33. Consequently, the average number of arcs meeting at each point xi ∈ X is 66/13 > 5, which contradicts the fact that at most ﬁve arcs meet at any of the v points. Case 2. f4 = 1. Then it follows from Euler’s formula that f3 = 20, e = 32, and 4 arcs meet at one point and 5 at every other. In fact, it is impossible to construct such a network. This can be veriﬁed by starting from the quadrilateral and adding triangles one by one. Hence we have proved that k(S3 ) ≤ 12.

(1.21)

1.4. Kepler’s Conjecture

13

Then, by (1.1), (1.17), and (1.21) Theorem 1.4 is proved.

2

1.4. Kepler’s Conjecture √ Routine computation yields that the packing density of S3 + Λ3 is π/ 18, which implies that π (1.22) δ ∗ (S3 ) ≥ √ . 18 Based on this fact and (1.2), J. Kepler in 1611 made the following conjecture. Kepler’s Conjecture.

π δ(S3 ) = √ . 18

By (1.4), if the volume of every Dirichlet-Voronoi cell of any packing √ S3 + X were greater than or equal to 4 2, one would be able to deduce π δ(S3 ) ≤ √ 18 and hence, by (1.22), π δ ∗ (S3 ) = δ(S3 ) = √ . 18 Unfortunately, the √ volumes of some Dirichlet-Voronoi cells of some packings are less than 4 2. For example, let D be one of the smallest regular dodecahedra circumscribed to S3 . Take X = {o, 2x1 , 2x2 , . . . , 2x12 } , where xi ∈ bd(S3 ) ∩ bd(D). Then, S3 + X is a packing (see Figure 1.4), and the Dirichlet-Voronoi cell D(o), deﬁned with respect to X, is the dodecahedron D itself. By routine computation, one obtains

√ 2π π v(D) = 20 1 − 2 cos tan < 4 2. 5 5 This example shows the fundamental diﬃculty of Kepler’s conjecture. Clearly, the density of any sphere packing may be improved by adding spheres as long as there is suﬃcient room to do so. When there is no longer room to add additional spheres we say that the sphere packing is

14

1. The Gregory-Newton Problem, Kepler’s Conjecture

saturated. Without loss of generality, we assume that the sphere packings considered in the following subsections are saturated. We now introduce two approaches to Kepler’s conjecture.

1.4.1. L. Fejes T´ oth’s Program and Hsiang’s Approach Consider the system of Dirichlet-Voronoi cells associated with √ a sphere packing. If the volume of a Dirichlet-Voronoi cell is less than 4 2, it seems √ likely that the volume of some of its neighbors will be larger than 4 2. Therefore, it is reasonable to consider a locally averaged density, which is what L. Fejes T´ oth [9] proposed in 1953 to attack Kepler’s conjecture. However, he was unable to realize this program. In 1993, Hsiang [2] announced a proof of Kepler’s conjecture. Unfortunately this has been found to contain errors and has not been accepted by experts (see Hales [3]). In fact, the fundamental ideas of both L. Fejes T´ oth’s program and Hsiang’s approach are the same. We only attempt to sketch their key ideas here. Let h be a ﬁxed number. Two spheres S3 + xi and S3 + xj in a given packing S3 + X are called h-neighbors of each other if xi , xj ≤ h. Take Xi = {x ∈ X : x, xi ≤ h} and, for convenience, enumerate it using double indices, namely setting Xi = {xij : j = 1, 2, . . . , mi } , where mi = card{Xi}. Then the h-locally averaged density of S3 +X around S3 + xi is deﬁned by

mi j=1 µij σ(xij , X)

mi σ(xi , X) = , j=1 µij where µij =

v(D(xij )) mi

and σ(xij , X) =

v(S3 ) . v(D(xij ))

Lemma 1.2 (Hsiang [2]). Let S3 + X be a saturated packing, and set µi =

mi j=1

µij

1.4. Kepler’s Conjecture

15

to be the weight assigned to S3 + xi . Then,

∈lI3 µi σ(xi , X) lim sup xi = δ(S3 , X). l→∞ xi ∈lI3 µi Proof. Since S3 + X is saturated, the diameter of any Dirichlet-Voronoi cell is bounded by 4 from above. Thus, by (1.4) and the deﬁnitions of µi and σ(xi , X) it follows that

µi σ(xi , X) =

xi ∈lI3

=

mi xi ∈lI3 j=1 mi xi ∈lI3 j=1

µij σ(xij , X) v(S3 + xij ) mi

= card{X ∩ lI3 }v(S3 ) + O l2 and xi ∈lI3

Hence,

lim sup

µi =

mi v(D(xij ))

mi 2 = v(lI3 ) + O l . xi ∈lI3 j=1

µi σ(xi , X) card{X ∩ lI3 }v(S3 ) = lim sup µ v(lI3 ) l→∞ xi ∈lI3 i

xi ∈lI3

l→∞

= δ(S3 , X). 2

Lemma 1.2 is proved. From Lemma 1.2 it follows that for any ﬁxed number h, δ(S3 ) ≤ sup sup σ(xi , X), X xi ∈X

where the outer supremum is over all sets X such that S3 + X is a packing. Thus, if it can be proved that for a suitable number h, π σ(xi , X) ≤ √ 18

(1.23)

holds for every point xi of any set X such that S3 +X is a packing, Kepler’s conjecture will follow. In this way, the original global problem has been converted into a local one. Since S3 + X is saturated, the number of combinatorial types of the Dirichlet-Voronoi cells is ﬁnite. Thus one can try to verify (1.23) by checking a ﬁnite number of cases. Clearly, the number of cases depends on the

16

1. The Gregory-Newton Problem, Kepler’s Conjecture

choice of h. In dealing with the individual cases linear programming plays a very important role. L. Fejes T´ oth [9] suggested h = 2.0523 . . . in his program. Hsiang [2] took h = 2.18 in his approach.

1.4.2. Delone Stars and Hales’ Approach Let X be a set of points such that S3 + X is a saturated packing. Then E 3 can be decomposed into simplices such that their vertices belong to X and the interior of any sphere circumscribing such a simplex contains no point of X. This decomposition is called the Delone triangulation, the simplices are called Delone simplices, and the union of the simplices with a common vertex x is called a Delone star (it is denoted by D (x), or simply D ). To attack Kepler’s conjecture by studying Delone stars, Hales [4] took the following approach. For convenience, we write

1 11π − 12 arccos √ , β= 3 3 √ −3π + 12 arccos(1/ 3) √ , γ= 8 and F (D ) = −v(D )γ + v (D ∩ (S3 + x)) . x∈X

Let D (x) be a Delone star with vertices x1 , x2 , . . ., xl , and denote the intersection of xxi with bd(S3 ) + x by pi . Joining pi and pj by a geodesic arc if and only if max {x, xi , x, xj , xi , xj } ≤ 2.51, we get a network on bd(S3 )+x that divides the surface into a set of regions, say A1 , A2 , . . ., Am . Let Vi be the cone with vertex x over Ai , and write Ci = Vi ∩ D (x). The Delone star is divided into m clusters C1 , C2 , . . ., Cm . Assign a score σ(Ci ) to every cluster, and deﬁne

σ(D (x)) =

m

σ(Ci )

i=1

(the exact score for an individual cluster is based on its structure and is very complicated). σ(D (x)) is called the score of D (x). It has the following properties.

1.4. Kepler’s Conjecture

17

1. The score of a cluster depends only on the cluster, and not on the way it sits in a Delone star or in the Delone triangulation of the space. 2. If D is a Delone star of the face-centered cubic packing or the hexagonal close packing of S3 , then σ(D ) = 8β. 3. For any saturated packing S3 + X, we have σ(D (x)) = F (D (x)) + O r2 . x∈X∩rS3

(1.24)

x∈X∩rS3

Lemma 1.3 (Hales [4]). Let S3 + X be a saturated packing. If max {σ(D (x))} = η ≤ x∈X

then δ(S3 , X) ≤

16π , 3

16πγ . 16π − 3η

In particular, if η = 8β, then π δ(S3 , X) ≤ √ . 18 Proof. Since S3 + X is a saturated packing, d(D (x)) ≤ 4 holds for every point x ∈ X. Thus, the number of points x ∈ X such that D (x) meets the boundary of rS3 has order O(r2 ). For convenience, we write m(r) = card {X ∩ rS3 } . Since the Delone stars give a fourfold cover of E 3 , by (1.24) we have

4π 4 −v(rS3 )γ + m(r) = − v(D (x))γ 3 x∈X∩rS3

+ v(D (x) ∩ (S3 + y)) + O r2 y∈X

=

x∈X∩rS3

=

F (D (x)) + O r2 σ(D (x)) + O r2

x∈X∩rS3

≤ m(r)η + O r2 . In other words,

m(r)

4π η − 3 4

≤ v(rS3 )γ + O r2 .

18 Thus,

1. The Gregory-Newton Problem, Kepler’s Conjecture

m(r)v(S3 ) 16πγ 1 ≤ +O . v(rS3 ) 16π − 3η r

The ﬁrst part of the lemma follows. Clearly, the second part is the special case η = 8β. Lemma 1.3 is proved. 2 In order to use this lemma to prove Kepler’s conjecture, Hales [4] proposed a program to verify the following assertions. Assertion 1.1. If all the regions are triangles, then σ(D ) ≤ 8β. Assertion 1.2. If the corresponding region of a cluster C has more than three sides, then σ(C ) ≤ 0. Assertion 1.3. If all the regions are triangles and quadrilaterals (excluding the case of pentagonal prisms), then σ(D ) ≤ 8β. Assertion 1.4. If one region has more than four sides, then σ(D ) ≤ 8β. Assertion 1.5. If D is a pentagonal prism, then σ(D ) ≤ 8β. Remark 1.4. A Delone star is called a pentagonal prism if its regions consist of ten triangles and ﬁve quadrilaterals, with the ﬁve quadrilaterals in a band around the equator, capped on both ends by ﬁve triangles. Proofs of these assertions were announced in Hales [4–7] and Ferguson [1], respectively. In fact, Hales [6, 7] and Ferguson [1] employed a diﬀerent decomposition and a diﬀerent score system formulated in Ferguson and Hales [1]. However, the basic idea is the same as Hales [4]. Their proofs are extraordinarily complicated. First a computer is used to list all the possibilities for the regions. Then optimization is used to deal with each case. In the course of this proof the computer plays a very important role.

1.5. Some General Remarks Kepler’s conjecture was made in 1611 based on the observation of S3 + Λ3 , while the Gregory-Newton problem was proposed in 1694 during a famous discussion. Over the course of the centuries, these problems and their generalizations have attracted the attention of many mathematicians. This section reviews related results that will not be discussed in other chapters. As with many other problems in geometry, the ﬁrst signiﬁcant progress concerning packing densities was made in E 2 . In 1773, by studying the

1.5. Some General Remarks

19

minimum of positive binary quadratic forms (this method will be discussed in Chapter 2), Lagrange [1] deduced π δ ∗ (S2 ) = √ . 12

(1.25)

In 1831, Gauss [1] used similar methods to prove a conjecture of Seeber [1] that implies π δ ∗ (S3 ) = √ . (1.26) 18 Besides the face-centered cubic lattice packing, for which this density is attained, Barlow [1] found inﬁnitely many nonlattice packings of spheres with the same density. They are the laminations of hexagonal layers of spheres. Removing the lattice restriction, the ﬁrst proof of π δ(S2 ) = √ 12

(1.27)

was achieved by Thue [1] and [2]. Roughly speaking, Thue’s idea was to compute the area left uncovered by the circular disks in certain ﬁnite packings. His method was developed further by Groemer [1] in 1960. Later, diﬀerent proofs of (1.27) were discovered by L. Fejes T´ oth [1], Segre and Mahler [1], Davenport [3], and Hsiang [1]. The proof of Theorem 1.3 is essentially that of L. Fejes T´ oth. Clearly (1.25), (1.26), and (1.27) all support Kepler’s conjecture. In 1900, at the International Congress of Mathematicians in Paris, D. Hilbert (see [1]) listed this conjecture as the third part of his 18th problem. Thus it became one of the most popular problems among mathematicians. C.A. Rogers once commented, “Many mathematicians believe, and all physicists √ know” that the density of the densest sphere packings in E3 is π/ 18. In 1976, in a review of Hilbert’s 18th problem, Milnor [1] wrote on this conjecture, “This is a scandalous situation since the (presumably) correct answer has been known since the time of Gauss (compare Hilbert and CohnVossen). All that is missing is a proof.” In 1929, Blichfeldt [1] obtained the ﬁrst upper bound 0.835 for δ(S3 ). This has been successively improved by Rankin [1], Rogers [10], Lindsey [1], and Muder [1] and [2] to 0.773055. In 1953, in his well-known book, L. Fejes T´ oth suggested a program to attack Kepler’s conjecture by checking a ﬁnite number of cases. In 1987, Dauenhauer and Zassenhaus [1] made a local approach and suggested the possibility of verifying this conjecture with the help of a computer. In 1993, Hsiang [2] announced a proof using a computer. Unfortunately, it has been found to contain errors (see Hales [3], Hsiang [3], and the preface of Conway and Sloane [1]). Recently, Hales [4–7] and Ferguson [1] announced another proof. To deal with the thousands of cases considered in their proof, even an eﬃcient computer has to run for years.

20

1. The Gregory-Newton Problem, Kepler’s Conjecture

H. Minkowski was the ﬁrst mathematician to study the packings of general convex bodies, followed by E. Hlawka, C.A. Rogers, L. Fejes T´ oth, and many others. In 1904, Minkowski [5] developed a method by which one can determine the value of δ∗ (K) for any three-dimensional convex body K. As examples he considered the cases of the octahedron and the tetrahedron. Unfortunately, he made a mistake in the tetrahedral case (see Groemer [3]). Later, this mistake was corrected by Hoylman [1]. Although Minkowski’s method is important in theory, it was considered not to be practical. However, recently, Betke and Henk [2] used Minkowski’s work as a starting point for an algorithm by which one can determine the value of δ∗ (K) for any three-dimensional polytope. As an application, they calculated the value of δ ∗ (K) for all regular and Archimedean polytopes. Around 1950, L. Fejes T´ oth [5] and Rogers [4] proved δ ∗ (K) = δ(K) oth for an arbitrary convex domain in E 2 . Thirty years later, L. Fejes T´ [12] presented an elegant new proof of this result. In high dimensions, it is commonly believed (see Rogers [14], Gruber and Lekkerkerker [1]) that there exist convex bodies K such that δ ∗ (K) < δ(K). Perhaps some high-dimensional spheres have this property. However, no example has been conﬁrmed as yet. As early as 1896, Minkowski [7] studied lattice packings of a general n-dimensional convex body K, and proved that k ∗ (K) ≤ 3n − 1, where equality holds if K is a parallelepiped. In 1957 Hadwiger [1] gave an elegant proof (see the proof of Theorem 1.1) of k(K) ≤ 3n − 1, improving the above result. A few years later, Groemer [2] produced a more detailed proof. On the other hand, it was conjectured by Gr¨ unbaum [1] that the best lower bound for k(K) is n(n + 1), which can be attained by n-dimensional simplices. In 1996, Zong [3] disproved the second part of this conjecture. Very recently, exponential lower bounds for k(K) were discovered by Talata [1] and Larman and Zong [1]. In the plane case, conﬁrming a conjecture of Hadwiger [1], Gr¨ unbaum [1] proved that 8 if K is a parallelogram, k(K) = k∗ (K) = (1.28) 6 otherwise. For a survey on the kissing numbers of general convex bodies we refer to Zong [7].

1.5. Some General Remarks

21

In the literature R. Hoppe has often been cited as the ﬁrst mathematician to solve the Gregory-Newton problem. In fact, his proof is not complete. In 1953, using arguments from graph theory, Sch¨ utte and van der Waerden [1] completely solved this problem, in favor of Newton. In 1956, Leech [1] gave a new proof. Very recently, Leech’s proof was further improved by Aigner and Ziegler [1].

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2. Positive Deﬁnite Quadratic Forms and Lattice Sphere Packings

2.1. Introduction There is a remarkable relationship between lattice sphere packings and positive deﬁnite quadratic forms. This relationship plays an important role in determining the values of δ∗ (Sn ) and k∗ (Sn ) for small n. Let Λ be a lattice with a basis {a1 , a2 , . . . , an }, where ai = (ai1 , ai2 , . . . , ain ), and write   a11 a12 · · · a1n  a21 a22 · · · a2n    A= . .. ..  . ..  .. . . .  an1 an2 · · · ann For convenience, we denote the n-dimensional integer lattice by Zn . There is a positive deﬁnite quadratic form related to the lattice Λ, Q(x) = xA, xA = xAA x , where A and x indicate the transposes of A and x, respectively. Let r(Λ) be the largest number such that r(Λ)Sn + Λ is a packing, let dis(Q) be the discriminant of the quadratic form Q(x), and write m(Q) =

min

z∈Zn \{o}

It is easy to see that r(Λ) =

1 2

Q(z).

m(Q)

24

2. Quadratic Forms, Lattice Sphere Packings

and dis(Q) = det(Λ)2 . Hence, δ(r(Λ)Sn , Λ) =

ωn r(Λ)n ωn m(Q)n/2 = . det(Λ) 2n dis(Q)

On the other hand, for each positive deﬁnite quadratic form Q(x) = xSx , where S is a positive deﬁnite symmetric matrix with entries sij , there is a corresponding lattice, Λ = {zA : z ∈ Zn }, such that S = AA . Therefore, there is also a corresponding lattice sphere packing r(Λ)Sn + Λ. Hence, letting F be the family of all positive deﬁnite quadratic forms in n variables, one has ωn m(Q)n/2 δ ∗ (Sn ) = sup . (2.1) Q(x)∈F 2n dis(Q) Similarly, letting M (Q) be the set of points z ∈ Zn \ {o} at which Q(z) attains its minimum, we have k ∗ (Sn ) = max {card{M (Q)}} . Q(x)∈F

(2.2)

Thus, the geometric problems of determining the values of δ∗ (Sn ) and k ∗ (Sn ) are reformulated as arithmetic ones. Write

m(Q) f (Q) = , n dis(Q)

and regard it as a function of the coeﬃcients of Q(x). Then Q(x) is called respectively a stable form or an absolutely stable form if f (Q) attains a local maximum or an absolute maximum at Q(x). If, in addition, m(Q) = 1, then Q(x) is called an extreme form or a critical form, respectively. Usually, the number m(Q) (2.3) γn = sup f (Q) = sup n dis(Q) Q(x)∈F Q(x)∈F is called Hermite’s constant. From (2.1) and (2.3) it follows that δ ∗ (Sn ) = ωn

γ n/2 n

4

.

(2.4)

Moreover, it is easy to see that the density of r(Λ)Sn + Λ is respectively a local maximum or an absolute maximum if and only if the corresponding form Q(x) is stable or absolutely stable. Therefore, the two problems, of determining the densest lattice sphere packings and determining the critical positive deﬁnite quadratic forms, are equivalent. Let U be a unimodular matrix and write Q(x) = xU SU x .

2.2. The Lagrange-Seeber-Minkowski Reduction

25

We say that Q(x) is equivalent to Q(x). Since the map z → zU is an automorphism in Zn , one has  = m(Q),   m(Q) = card{M (Q)}, (2.5) card{M (Q)}   dis(Q) = det(U SU ) = dis(Q). Let Q be the subfamily of positive deﬁnite quadratic forms that are equivalent to Q(x). In other words, Q = {xU SU x : U is a unimodular matrix} . Then, the family F can be represented as a union of diﬀerent subfamilies Q. So, by (2.5), if in each subfamily Q a particular form can be chosen, the problem of determining the values of δ∗ (Sn ), k∗ (Sn ), and γn using (2.1), (2.2), and (2.3) can be simpliﬁed. This is the basic idea of reduction theory. In this chapter the values of δ∗ (Sn ), k∗ (Sn ), and γn for small n will be determined using (2.1), (2.2), and (2.3) together with the assistance of diﬀerent reductions.

2.2. The Lagrange-Seeber-Minkowski Reduction and a Theorem of Gauss Deﬁnition 2.1. As usual, we denote the greatest common divisor of k integers z1 , z2 , . . . , zk by (z1 , z2 , . . . , zk ) (here we use bold parentheses). A positive deﬁnite quadratic form Q(x) = xSx is said to be L-S-M reduced if s1j ≥ 0,

j = 2, 3, . . . , n,

and Q(z) ≥ sii

(2.6)

for all integer vectors z = (z1 , z2 , . . . , zn ) such that (zi , zi+1 , . . . , zn ) = 1,

i = 1, 2, . . . , n.

Lemma 2.1 (Minkowski [6]). Every positive deﬁnite quadratic form is equivalent to an L-S-M reduced one. Proof. Let Q(x) = xSx be a positive deﬁnite quadratic form. Using basic algebra we know that there exists a nonsingular matrix A such that S = AA . Denote the lattice {zA : z ∈ Zn } by Λ. Then Q(z)1/2 is the Euclidean norm of the vector zA in Λ. Hence, Lemma 2.1 can be proved by showing that the following assertion holds.

26

2. Quadratic Forms, Lattice Sphere Packings

Assertion 2.1. Every lattice Λ has a basis {a1 , a2 , . . . , an } such that n j=1

zj aj ,

n

≥ ai , ai

zj aj

(2.7)

j=1

for all integer vectors z = (z1 , z2 , . . . , zn ) with (zi , zi+1 , . . . , zn ) = 1,

i = 1, 2, . . . , n,

and a1 , aj ≥ 0,

j = 2, 3, . . . , n.

(2.8)

We now prove this assertion. First, we take a1 to be a point of Λ satisfying a1 , a1 = min a, a. a∈Λ\{o}

Let B2 be the set of points a ∈ Λ such that {a1 , a} can be extended to a basis for Λ. From linear algebra we know that the set B2 is nonempty. Then, take a2 to be a point of B2 satisfying a1 , a2 ≥ 0 and a2 , a2 = min a, a. a∈B2

Inductively, assume that i < n and a1 , a2 , . . . , ai have been chosen. Let Bi+1 be the set of points a ∈ Λ such that {a1 , a2 , . . . , ai , a} can be completed to a basis for Λ. Note that again we know that Bi+1 is nonempty. Take ai+1 to be a point of Bi+1 satisfying a1 , ai+1 ≥ 0 and ai+1 , ai+1 = min a, a. a∈Bi+1

(2.9)

It is easy

n to see that {a1 , a2 , . . . , an } is a basis of Λ that satisﬁes (2.8). Let a = j=1 zj aj be a point of Λ such that (zi , zi+1 , . . . , zn ) = 1. From linear algebra we know that {a1 , a2 , . . . , ai , a} can be extended to a basis for Λ. In other words, a ∈ Bi+1 . Then, (2.7) follows from (2.9). Therefore, Assertion 2.1 and Lemma 2.1 are proved. 2 In fact, by a theorem of Minkowski [6], in order to verify that a positive deﬁnite quadratic form is L-S-M reduced one only need check that (2.6) holds for a ﬁnite number of integer vectors. Here we introduce two practical criteria for the L-S-M reduced binary forms and the L-S-M reduced ternary forms. Lemma 2.2 (Lagrange [1]). A positive deﬁnite binary quadratic form Q(x) = xSx is L-S-M reduced if and only if s11 ≤ s22 , (2.10) 0 ≤ 2s12 ≤ s11 .

2.2. The Lagrange-Seeber-Minkowski Reduction

27

Proof. Let Q(x) be a form satisfying (2.10). Then, for z = (z1 , z2 ) ∈ Z2 , Q(z) = s11 (z1 )2 + 2s12 z1 z2 + s22 (z2 )2 ≥ s11 (z1 )2 − s11 |z1 z2 | + s11 (z2 )2 + (s22 − s11 )(z2 )2 ! = s11 (z1 )2 − |z1 z2 | + (z2 )2 + (s22 − s11 )(z2 )2 . Clearly, for z = o,

(z1 )2 − |z1 z2 | + (z2 )2 ≥ 1.

Therefore, since s22 ≥ s11 , we have s11 , if z = o, Q(z) ≥ s22 , if z2 = 0. Then, it follows from Deﬁnition 2.1 that Q(x) is L-S-M reduced. On the other hand, if Q(x) is L-S-M reduced, the inequalities of (2.10) follow from Q(1, 0) ≤ Q(0, 1) and Q(0, 1) ≤ Q(1, −1). Hence, Lemma 2.2 is proved. 2 Remark 2.1. Let Q(x) be an L-S-M reduced positive deﬁnite binary quadratic form. By (2.10), we have dis(Q) s11 s22 − (s12 )2 s11 s22 − (s11 )2 /4 = ≥ s11 s22 s11 s22 s11 s22 s11 3 ≥ 1− ≥ . 4s22 4 Hence, m(Q) = s11 ≤

√

s11 s22 ≤

"

4 3 dis(Q).

Then, by (2.1) and certain example it follows that π δ ∗ (S2 ) = √ . 12 Thus we obtain an arithmetic proof for the lattice case of Theorem 1.3. Examining the above argument it is easy to see that every critical positive deﬁnite binary quadratic form is equivalent to Q(x) = (x1 )2 + x1 x2 + (x2 )2 , which implies the uniqueness of the densest lattice circle packing. This idea goes back to J.L. Lagrange. Lemma 2.3 (Seeber [1]). A positive deﬁnite ternary quadratic form Q(x) = xSx is L-S-M reduced if and only if  s11 ≤ s22 ≤ s33 ,      0 ≤ 2s12 ≤ s11 , 0 ≤ 2s13 ≤ s11 , (2.11)   0 ≤ 2|s | ≤ s ,  23 22   −2s23 ≤ s11 + s22 − 2(s12 + s13 ).

28

2. Quadratic Forms, Lattice Sphere Packings

Proof. For convenience, let (2.11.i) indicate the ith inequality of (2.11). Assume that Q(x) satisﬁes (2.11), we proceed to show that it is L-S-M reduced. In other words,   s11 , if (z1 , z2 , z3 )= 1, s22 , if (z2 , z3 )= 1, (2.12) Q(z) ≥  s33 , if z3 = 0. If z3 = 0, then by (2.11.1) and (2.11.2), Q(z1 , z2 , 0) is L-S-M reduced. Then, the ﬁrst two inequalities of (2.12) follow from Lemma 2.2. Now we proceed to show the last inequality of (2.12) by dealing with two cases. Case 1. z3 = 0 and z1 = 0. Deﬁne θ = 1 if s23 ≥ 0, and θ = −1 if s23 < 0. From (2.11.1) and (2.11.4) it follows that Q(0, z2 , θz3 ) is L-S-M reduced. Thus, by Lemma 2.2, Q(0, z2 , z3 ) ≥ s33 . Similarly, it can be proved that Q(z1 , 0, z3 ) ≥ s33 . Case 2. z1 z2 z3 = 0. Routine computation yields that Q(z) = (s11 − s12 − s13 )(z1 )2 + (s22 − s12 − |s23 |)(z2 )2 + (s33 − s13 − |s23 |)(z3 )2 + Σ,

(2.13)

where Σ = s12 (z1 + z2 )2 + s13 (z1 + z3 )2 + |s23 |(z2 + θz3 )2 . From the ﬁrst four inequalities of (2.11) it follows that all the coeﬃcients of (2.13) and Σ are nonnegative. Assume that s12 ≤ s13 ≤ |s23 |. We consider three subcases. i. At least two of the three terms z1 + z2 , z1 + z3 , and z2 + θz3 are nonzero. Then Σ ≥ 2s12 , and therefore, by (2.13), (2.11.3), and (2.11.4), Q(z) ≥ s11 + s22 + s33 − 2(s12 + s13 + |s23 |) + 2s12 = (s11 − 2s13 ) + (s22 − 2|s23 |) + s33 ≥ s33 . ii. Exactly two of the three terms z1 + z2 , z1 + z3 , and z2 + θz3 are zero. Then, the third term must be a multiple of two, and therefore Σ ≥ 2s12 . Similarly to the previous subcase we have Q(z) ≥ s33 . iii. z1 + z2 = z1 + z3 = z2 + θz3 = 0. Then θ = −1 and s23 < 0. Thus, by (2.13) and (2.11.5), Q(z) = s11 + s22 + s33 − 2(s12 + s13 − s23 ) = s33 + 2s23 + [s11 + s22 − 2(s12 + s13 )] ≥ s33 . On the other hand, if Q(x) is an L-S-M reduced ternary form, then (2.11) follows from the inequalities Q(1, 0, 0) ≤ Q(0, 1, 0) ≤ Q(0, 0, 1),

2.2. The Lagrange-Seeber-Minkowski Reduction

29

Q(0, 1, 0) ≤ Q(1, −1, 0), Q(0, 0, 1) ≤ Q(1, 0, −1), Q(0, 0, 1) ≤ Q(0, 1, ±1), and Q(0, 0, 1) ≤ Q(−1, 1, 1). Hence, Lemma 2.3 is proved. 2 Theorem 2.1 (Gauss [1]). π δ∗ (S3 ) = √ . 18

(2.14)

Furthermore, the densest lattice packing of S3 is unique up to rotation and reﬂection. Proof. By (2.1), (2.5), Lemma 2.1, and Lemma 2.3, in order to prove (2.14) it is suﬃcient to show that dis(Q) ≥ 12 s11 s22 s33

(2.15)

for every positive deﬁnite ternary quadratic form Q(x) = xSx that satisﬁes (2.11). It is well-known that dis(Q) = s11 s22 s33 + 2s12 s13 s23 − s11 (s23 )2 − s22 (s13 )2 − s33 (s12 )2 . (2.16) Fixing s11 , s22 , and s33 , and regarding dis(Q) = f (s12 , s13 , s23 ) as a function of s12 , s13 , and s23 , we proceed to prove (2.15) by dealing with two cases. Case 1. s23 < 0. Since dis(Q) is a convex function of s23 , its minimum, under the constraints of (2.11), is attained at s23 = min s12 + s13 − 12 (s11 + s22 ), − 12 s22 . Thus, this case can be dealt with in two subcases. i. s23 = s12 + s13 − (s11 + s22 )/2. Then, it follows from (2.16) that dis(Q) = a1 (s13 )2 + b1 s13 + c1 ,

(2.17)

where a1 , b1 , and c1 are independent of s13 , and a1 = −[(s11 − 2s12 ) + s22 ] < 0. In addition, by (2.11.4) and the assumption of this subcase, s11 − 2s12 ≤ 2s13 .

(2.18)

By (2.11.3) the minimal value of (2.17) is attained either at s13 = 0 or at s13 = s11 /2. When s13 = 0, it follows from (2.11.2), (2.18), and the

30

2. Quadratic Forms, Lattice Sphere Packings

assumption of this subcase that s12 = s11 /2 and s23 = −s22 /2, which will be dealt with in the following subcase. When s13 = s11 /2 we have dis(Q) = a2 (s12 )2 + b2 s12 + c2 ,

(2.19)

where a2 , b2 , and c2 are independent of s12 , and a2 = −s33 < 0. By (2.11.2) the minimal value of (2.19) is attained either at s12 = 0 or at s12 = s11 /2. Then, since s11 ≤ s22 ≤ s33 , routine computation yields   s11 s22 s33 1 − s11 − s22 ≥ 1 s11 s22 s33 , if s12 = 0, 4s33 4s33 2 dis(Q) =  s11 s22 s33 1 − s11 − s22 ≥ 1 s11 s22 s33 , if s12 = 1 s11 . 4s22 4s33 2 2 ii. s23 = −s22 /2. In this case, it follows from (2.16) that dis(Q) = s11 s22 s33 − s22 s12 s13 − 14 s11 (s22 )2 − s22 (s13 )2 − s33 (s12 )2 . (2.20) Also, by (2.11.5), 2s13 ≤ s11 − 2s12 .

(2.21)

We calculate the derivative ∂dis(Q) = −s22 (s12 + 2s13 ) ≤ 0. ∂s13 Hence, in order to minimize the value of dis(Q) we shall choose the maximum possible value for s13 under the conditions 2s13 ≤ s11 and (2.21), namely s13 = s11 /2 − s12 . Then, it follows from (2.20) that dis(Q) = a3 (s12 )2 + b3 s12 + c3 , where a3 , b3 , and c3 are independent of s12 , and a3 = −s33 < 0. Thus by a similar argument to the previous subcase we obtain dis(Q) ≥ 12 s11 s22 s33 . Case 2. s23 ≥ 0. Then, the last condition in (2.11) follows from the others. Arguing as before, it follows that the minimal value of dis(Q) will be attained either at s23 = 0 or at s23 = s22 /2. Hence, by (2.16), dis(Q) ≥ min{f (s12 , s13 , 0), f (s12 , s13 , s22 /2)} s s 11 22 = s11 s22 s33 − s22 − s12 s13 4 − s22 (s13 )2 − s33 (s12 )2 . Considering the right-hand side of this inequality as a function of s12 and s13 , and denoting it by g(s12 , s13 ), simple analysis yields dis(Q) ≥

min

0≤s12 ≤s11 /2, 0≤s13 ≤s22 /2

g(s12 , s13 ) = g(s11 /2, s22 /2)

s11 s22 = s11 s22 s33 1 − − 4s22 4s33 1 ≥ 2 s11 s22 s33 .

2.3. Mordell’s Inequality on Hermite’s Constants

31

Hence (2.15) follows from Case 1 and Case 2. Also, tracing back through the above argument, it can be veriﬁed that every critical positive deﬁnite ternary quadratic form is equivalent to Q(x) = (x1 )2 + (x2 )2 + (x3 )2 + x1 x2 + x1 x3 + x2 x3 , which implies the second part of our theorem. Theorem 2.1 is proved.

2

2.3. Mordell’s Inequality on Hermite’s Constants and a Theorem of Korkin and Zolotarev Lemma 2.4 (Mordell [1]). n−1

γn ≤ (γn−1 ) n−2 . Proof. Let Xi =

n

aij xj ,

i = 1, 2, . . . , n,

(2.22)

j=1

be n linear forms with real coeﬃcients and determinant 1, and let bij be the cofactor of aij of A, the n × n matrix with entries aij . Deﬁne n linear forms Y1 , Y2 , . . . , Yn in variables y1 , y2 , . . . , yn by means of the identity n

Xi Yi =

i=1

For example, Yi =

n

n

xi yi .

(2.23)

i=1

bij yj ,

i = 1, 2, . . . , n.

j=1

Obviously, the n linear forms Yi also have determinant 1. Letting Q(x) = xSx = xAA x be a positive deﬁnite quadratic form with dis(Q) = 1, we can write 2

2

Q(x) = (X1 ) + (X2 ) + · · · + (Xn )

2

with appropriate linear forms Xi satisfying (2.22). Then let Q∗ (y) = (Y1 ) + (Y2 ) + · · · + (Yn ) 2

2

2

(2.24)

32

2. Quadratic Forms, Lattice Sphere Packings

be the form associated with Q(x) using (2.23). By applying a unimodular substitution in (2.24) and changing Q∗ (y) into an equivalent form, we may assume that the minimum m(Q∗ ) is given by (y1 , y2 , . . . , yn ) = (0, 0, . . . , 1). In other words, m(Q∗ ) =

n

2

(bin ) .

(2.25)

i=1

In this case we must then also

n apply a corresponding unimodular substitution to the xi s such that i=1 xi yi remains unaltered, and then Q(x) is replaced by an equivalent form. Now we estimate the values of Q(z) for zn = 0. In this case, Q(z) becomes a quadratic form in n − 1 variables, namely

2 n n−1 Q (z) = aij zj . i=1

j=1

For convenience, we denote by S the (n − 1) × (n − 1) matrix with entries sij =

n

aki akj

k=1

for 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n − 1, and denote by M the n × (n − 1) matrix with entries aij for 1 ≤ i ≤ n and 1 ≤ j ≤ n − 1. Then, by (2.25), we have dis(Q ) = det(S ) = det(M M ) =

n

(bin )2

i=1

= m(Q∗ ) ≤ γn , and therefore 1

1

m(Q) ≤ m(Q ) ≤ γn−1 dis(Q ) n−1 ≤ γn−1 (γn ) n−1 . Since the right-hand side is independent of the form Q(x), it follows from (2.3) that 1 γn ≤ γn−1 (γn ) n−1 and consequently n−1

γn ≤ (γn−1 ) n−2 . Lemma 2.4 is proved.

2

From (2.4), Lemma 2.4, Theorem 2.1, and their proofs it is easy to deduce the density of the densest lattice sphere packings in E4 , which was determined by Korkin and Zolotarev by a much more complicated method.

2.4. Perfect Forms, Voronoi’s Method

33

Theorem 2.2 (Korkin and Zolotarev [1]). δ ∗ (S4 ) =

π2 . 16

Also, the densest lattice packing of S4 is unique up to rotation and reﬂection.

2.4. Perfect Forms, Voronoi’s Method, and a Theorem of Korkin and Zolotarev To determine the value of δ∗ (Sn ) by studying extreme forms, we introduce the concept of a perfect form. A positive deﬁnite quadratic form Q(x) is called perfect if it is determined uniquely by the equations Q(zi ) = m(Q). Extreme forms and perfect forms have the following important relationship. Lemma 2.5 (Korkin and Zolotarev [2]). Every extreme positive deﬁnite quadratic form is perfect. Proof. Let Q(x) = xSx be an extreme positive deﬁnite quadratic form in n variables, and assume that M (Q) = {±z1 , ±z2 , . . . , ±zk } . If, on the contrary, Q(x) is not perfect, there is a nonzero symmetric matrix S ∗ such that zi S ∗ zi = 0, i = 1, 2, . . . , k. Deﬁne

Qλ (x) = x(S + λS∗ )x .

It is easy to see that for a suitable positive number α, Qλ (x) is positive deﬁnite and m(Qλ ) = m(Q) = 1 (2.26) whenever |λ| ≤ α. It is well-known from linear algebra that there is an orthogonal matrix U such that both U SU and U S ∗ U are diagonal, with diagonal entries si and s∗i , respectively. Hence, dis(Qλ ) =

n # (si + λs∗i ). i=1

Writing f (λ) = log dis(Qλ ),

34

2. Quadratic Forms, Lattice Sphere Packings

simple analysis yields that f (λ) is a strictly concave function of λ. Therefore, one has (2.27) dis(Qλ ) < dis(Q) for λ > 0, or for λ < 0. Then, (2.26) and (2.27) together yield, m(Q) m(Qλ ) > , n n dis(Qλ ) dis(Q) which contradicts the assumption that Q(x) is extreme. Lemma 2.5 is proved. 2 Using Lemma 2.5, we can determine all the extreme forms if we can determine all the perfect forms. For this reason, since a perfect form can be determined by its minimum integer points, Voronoi [1] developed the following method. The method is very elegant. However, its proof is very complicated. So, here we only introduce the method itself. Voronoi’s Method. For convenience, we write d=

n(n + 1) . 2

Let Q1 (x) be a perfect form in n variables with m(Q1 ) = 1 and assume that M (Q1 ) = {±z1 , ±z2 , . . . , ±zk } , where zi = (zi1 , zi2 , . . . , zin ). Then, corresponding to zi and Q1 (x), respectively, in E d we introduce a point z∗i = (zi1 zi1 , zi1 zi2 , . . . , zi1 zin , zi2 zi2 , zi2 zi3 , . . . , zi2 zin , . . . , zin zin ) and a polyhedral convex cone V1 =

k

αi z∗i : αi ≥ 0 .

i=1

Assume that V1 has l1 − 1 facets Fi , i = 2, 3, . . . , l1 , which are determined by o∗ , z∗i1 , z∗i2 , . . . , z∗if (i) . Let Q∗i (x) be nonzero quadratic forms in n variables such that Q∗i (zij ) = 0, Deﬁning λi =

j = 1, 2, . . . , f (i).

Q1 (z) − 1 z∈Zn , Qi (z)<0 −Q∗ i (z) min ∗

2.4. Perfect Forms, Voronoi’s Method

35

and Qi (x) = Q1 (x) + λi Q∗i (x) for i = 2, 3, . . . , l1 , then Qi (x) are new perfect forms such that m(Qi ) = 1 and the corresponding polyhedral cone Vi joins V1 at the whole facet Fi . For convenience, we say that Qi (x) is a neighbor of Q1 (x). Repeating this process for Q2 (x), Q3 (x), . . . , Ql1 (x) and their neighbors, a sequence of perfect forms Q1 (x), Q2 (x), . . . , Ql (x)

(2.28)

can be found such that every neighbor of any of these forms is equivalent to one of them. Then, (2.28) is a complete system of perfect positive deﬁnite quadratic forms in n variables. In other words, every perfect form in n variables is equivalent to one of the l forms in (2.28). By this method, one can prove the following result with routine argument. Lemma 2.6 (Korkin and Zolotarev [3]). Let Un (x) =

n ≥ 2,

xi xj ,

1≤i≤j≤n

Vn (x) = Un (x) − x1 x2 ,

n ≥ 4,

and W5 (x) =

5 5 1 1 (xi )2 − x1 xi + 2 i=2 2 i=1

2≤i<j≤4

xi xj −

4

xi x5 .

i=2

For n ≤ 5, every perfect positive deﬁnite quadratic form Q(x) with m(Q) = 1 is equivalent to one of the seven forms U2 (x), U3 (x), U4 (x), V4 (x), U5 (x), V5 (x), or W5 (x). Then, the density of the densest lattice sphere packings in E5 can be deduced from (2.3), (2.4), Lemma 2.5, and Lemma 2.6. Theorem 2.3 (Korkin and Zolotarev [3]). δ ∗ (S5 ) =

π2 √ . 15 2

In addition, the densest lattice packing of S5 is unique up to rotation and reﬂection.

36

2. Quadratic Forms, Lattice Sphere Packings

2.5. The Korkin-Zolotarev Reduction and Theorems of Blichfeldt, Barnes, and Vet˘cinkin Deﬁnition 2.2. A positive deﬁnite quadratic form Q(x) is said to be K-Z reduced if

2 n n ci xi + tij xj , Q(x) = i=1

where |tij | ≤ ci =

1 2

j=i+1

and n

min

(zi ,zi+1 ,...,zn )∈Zn−i+1 \{o}

2 n cj zj + tjk zk .

j=i

k=j+1

Lemma 2.7 (Korkin and Zolotarev [2]). Every positive deﬁnite quadratic form is equivalent to a K-Z reduced form. Proof. Let Q(x) = xSx be a positive deﬁnite quadratic form in n variables, let A be an n × n matrix such that S = AA , and let Λ be the lattice {zA : z ∈ Zn }. Now we proceed to show that the lattice Λ has a special basis {b1 , b2 , . . . , bn } that corresponds to a K-Z reduced form.

b3 b2

p3 o

p2

b1

Figure 2.1 To seek such a basis eﬃciently, we denote the i-dimensional space gener i ated by {b1 , b2 , . . . , bi } by Hi , the parallelepiped { j=1 λj bj : |λj | ≤ 12 } by Pi , the orthogonal projection of bi to Hi−1 by pi (see Figure 2.1), and, as usual, the distance between a point x and a set X by d(x, X). Writing qi = bi − pi and Pi∗ =

i j=1

λj qj : |λj | ≤

(2.29) 1 , 2

2.5. The Korkin-Zolotarev Reduction it is easy to see that

37

vi (Pi ) = vi (Pi∗ ),

(2.30)

where vi (X) indicates the i-dimensional volume of X. First, take b1 to be a point of the lattice Λ such that b1 , b1 =

min a, a.

a∈Λ\{o}

Then, choose b2 to be a point of Λ such that d(b2 , H1 ) = min d(a, H1 ) a∈Λ\H1

and

p2 ∈ P1∗ .

(2.31)

Inductively, if b1 , b2 , . . . , bi have been ﬁxed, we choose bi+1 to be a point of Λ such that d(bi+1 , Hi ) = min d(a, Hi ) (2.32) a∈Λ\Hi

and

pi+1 ∈ Pi∗

(2.33)

((2.31) and (2.33) are guaranteed by the periodic behavior of the sublattice

{ ij=1 zj bj : zj ∈ Z} and (2.30)). Thus, we obtain a set of n points {b1 , b2 , . . . , bn } that is a basis for Λ and a corresponding set of n points {q1 , q2 , . . . , qn } that is an orthogonal basis for E n . Therefore, every point x of E n can be represented as x=

n

xi bi =

i=1

n

yi qi ,

(2.34)

i=1

where, by (2.29) and (2.33), yi = xi +

n

tij xj

(2.35)

j=i+1

and |tij | ≤ 12 .

(2.36)

Since {q1 , q2 , . . . , qn } is an orthogonal basis of E n , by (2.34) and (2.35) the form Q(x) is equivalent to n n n n ∗ Q (x) = xi bi , xi bi = yi qi , yi qi i=1

i=1

i=1

i=1

2 n n n = qi , qi (yi )2 = qi , qi xi + tij xj . i=1

i=1

j=i+1

38

2. Quadratic Forms, Lattice Sphere Packings

By (2.36) and (2.32) it is easy to see that Q∗ (x) is K-Z reduced with ci = qi , qi . Lemma 2.7 is proved. 2 If Q(x) is K-Z reduced, then m(Q) = c1

and dis(Q) = c1 c2 · · · cn .

Thus, letting R denote the family of K-Z reduced positive deﬁnite quadratic forms in n variables, by (2.1), (2.3), (2.5), and Lemma 2.7 it follows that δ ∗ (Sn ) = sup Q(x)∈R

and γn = sup Q(x)∈R

ωn (c1 )n/2 √ c1 c2 · · · cn

2n

√ n

c1 . c1 c2 · · · cn

(2.37)

(2.38)

Therefore, the values of δ∗ (Sn ) and γn can be determined by studying √ the relationship between c1 and n c1 c2 · · · cn for the K-Z reduced positive deﬁnite quadratic forms in n variables. Lemma 2.8 (Korkin and Zolotarev [2]). The outer coeﬃcients of a K-Z reduced positive deﬁnite quadratic form satisfy ci+1 ≥ 34 ci

f or i = 1, 2, . . . , n − 1,

(2.39)

ci+2 ≥ 23 ci

f or i = 1, 2, . . . , n − 2.

(2.40)

and

Proof. For convenience, we assume that Q(x) is a K-Z reduced ternary form with c1 = 1. By Deﬁnition 2.2, taking x = (0, 1, 0), (0, 0, 1), (−1, 1, 1), and (0, 1, −1), respectively, it follows that c2 + (t12 )2 ≥ 1,

(2.41)

c3 + c2 (t23 )2 + (t13 )2 ≥ 1,

(2.42)

c3 + c2 (1 + t23 ) + (1 − t12 − t13 ) ≥ 1,

(2.43)

c3 + c2 (1 − t23 )2 + (t12 − t13 )2 ≥ 1.

(2.44)

2

2

and Similarly, taking x = (0, 0, 1) and comparing Q(x) with c2 , one has c3 + c2 (t23 )2 ≥ c2 .

(2.45)

First, since |tij | ≤ 12 , it follows from (2.41) that c2 ≥ 1 − (t12 )2 ≥ 34 ,

(2.46)

2.5. The Korkin-Zolotarev Reduction

39

which implies (2.39). Now we proceed to prove (2.40) by dealing with three cases. Case 1. |t23 | ≤ 13 . Then, (2.45) and (2.46) imply c3 ≥ c2 1 − (t23 )2 ≥ 23 . Case 2. − 21 ≤ t23 < − 13 . By (2.41) and (2.45) it follows that c3 ≥ c2 1 − (t23 )2 ≥ 1 − (t12 )2 1 − (t23 )2 and therefore (t12 )2 ≥ 1 −

c3 . 1 − (t23 )2

(2.47)

Similarly, by (2.42) and (2.45) it follows that (t13 )2 ≥ 1 −

c3 . 1 − (t23 )2

(2.48)

Without loss of generality, we assume that both t12 and t13 are positive. Hence, by (2.43), (2.45), (2.47), and (2.48) we have c3 +

2 $ 1 + t23 c3 c3 + 1 − 2 1 − ≥ 1. 1 − t23 1 − (t23 )2

Then, routine analysis yields c3 ≥ 23 . Case 3. we have

1 3

< t23 ≤ 12 . As in Case 2, by (2.44), (2.45), (2.36), and (2.48) 1 − t23 c3 + c3 + 1 + t23

1 − 2

$ 1−

c3 1 − (t23 )2

2 ≥ 1.

Then, routine analysis yields c3 ≥ 23 . Thus, (2.40) is proved and hence the proof is complete.

2

Remark 2.2. The values of δ∗ (S2 ), δ∗ (S3 ), and δ∗ (S4 ) can be directly deduced from (2.37) and Lemma 2.8. Thus, we obtain diﬀerent proofs for Theorem 2.1 and Theorem 2.2. Similarly, one can determine the values of γ2 , γ3 , and γ4 by (2.38) and Lemma 2.8. Besides Lemma 2.8, there is another basic result about the outer coefﬁcients of a K-Z reduced positive deﬁnite quadratic form in n variables. Lemma 2.9 (Vet˘cinkin [2]). ci+1 ci+2 · · · ci+j ≥ where i + j ≤ n.

(ci )j , √ j+1 γ j+1

40

2. Quadratic Forms, Lattice Sphere Packings

Proof. Assume that n

Q(x) =

2 n ck xk + tkl xl

k=1

l=k+1

is K-Z reduced. Let Qij (x) =

i+j

2 i+j ck xk + tkl xl .

k=i

l=k+1

Then Qij (x) is a positive deﬁnite quadratic form in j +1 variables satisfying m(Qij ) = ci and dis(Qij ) = ci ci+1 · · · ci+j . Then, by the deﬁnition of Hermite’s constant it follows that √

j+1

ci = ci ci+1 · · · ci+j

m(Qij ) ≤ γj+1 , dis(Qij )

j+1

2

which proves Lemma 2.9.

Using (2.37), Lemma 2.7, Lemma 2.8, and Lemma 2.9, one can determine the densities of the densest lattice sphere packings in E6 , E 7 , and E 8 by considering many diﬀerent cases. Since it is impossible to relate the details in a few pages, we simply quote the results here. Theorem 2.4 (Blichfeldt [2]). δ ∗ (S6 ) =

π3 √ , 48 3

δ ∗ (S7 ) =

π3 , 105

and

δ∗ (S8 ) =

π4 . 384

Theorem 2.5 (Barnes [4] and Vet˘cinkin [2]). The densest lattice packing of Sn , n = 6, 7, or 8, is unique up to rotation and reﬂection. Remark 2.3. By Lemma 2.4, one can deduce the value of δ∗ (S8 ) from that of δ∗ (S7 ). Remark 2.4. Besides L-S-M reduction and K-Z reduction, there are several others such as Hermite’s reduction and Selling’s reduction that have been studied by Aﬄerbach, Babai, Baranovskii, Delone, Lenstra, Lenstra Jr., Lov´ asz, Ry˘ s kov, S˘ togrin, Tammela, Venkov, van der Waerden, and many others. Unfortunately, when n is comparatively large, none of these reductions are eﬃcient enough to deal with either δ∗ (Sn ) or k∗ (Sn ).

2.6. Perfect Forms, Lattice Kissing Numbers of Spheres

41

2.6. Perfect Forms, the Lattice Kissing Numbers of Spheres, and Watson’s Theorem To determine the lattice kissing numbers of spheres, we introduce another important result concerning perfect forms. Lemma 2.10 (Voronoi [2]). For every positive deﬁnite quadratic form Q(x) there is a perfect form Q (x) such that M (Q) ⊆ M (Q ). Proof. Let Q(x) = xSx be an imperfect positive deﬁnite quadratic form in n variables and assume that M (Q) = {±z1 , ±z2 , . . . , ±zk } . Since Q(x) is imperfect, there is a nonzero quadratic form Q∗ (x) = xS ∗ x such that Q∗ (zi ) = zi S ∗ zi = 0, i = 1, 2, . . . , k. Let Q1 be the family of all positive deﬁnite quadratic forms Qλ (x) = Q(x) + λQ∗ (x) such that m(Qλ ) = m(Q). For convenience, we write Z ∗ = Zn \ {M (Q) ∪ {o}}. Clearly, we have

min

Qλ ∈Q1

min∗ Qλ (z) = m(Q).

z∈Z

Thus, choosing a form Q1 (x) ∈ Q1 such that min Q1 (z) = m(Q),

z∈Z ∗

it is obvious that M (Q) ⊂ M (Q1 ). If Q1 (x) is perfect, the assertion of Lemma 2.10 is proved by taking Q (x) = Q1 (x). Otherwise, replacing Q(x) by Q1 (x) (or its successors) and repeating this process, one can obtain a sequence of positive deﬁnite quadratic forms Q(x), Q1 (x), Q2 (x), . . . (2.49) such that m(Qi ) = m(Q),

i = 1, 2, . . . ,

42

2. Quadratic Forms, Lattice Sphere Packings

and M (Q) ⊂ M (Q1 ) ⊂ M (Q2 ) ⊂ · · · .

(2.50)

On the other hand, since k∗ (Sn ) is bounded from both sides, the sequence (2.49) terminates at a certain form, say Ql (x). This form is perfect. Hence, taking Q (x) = Ql (x), Lemma 2.10 is proved. 2 By (2.2) and this lemma, we can determine the value of k∗ (Sn ) if we can ﬁnd all perfect forms in n variables. However, complete sets of positive deﬁnite perfect forms are known only when n ≤ 6 (see Lemma 2.6 and Barnes [4]). Using Lemma 2.10 and considering many complicated cases, G.L. Watson was able to prove the following theorem. Theorem 2.6 (Watson [4]). n

4

5

6

7

8

9

k (Sn )

24

40

72

126

240

272

∗

In addition, in each of these dimensions, the lattice packing of Sn at which k ∗ (Sn ) is attained is unique up to rotation and reﬂection. Remark 2.5. Let Dn and En be the lattices deﬁned in Chapter 9 and let Dn if 3 ≤ n ≤ 5, Λn = En if 6 ≤ n ≤ 8. It is easy to verify that both δ∗ (Sn ) and k∗ (Sn ) can be realized at Sn + Λn when 3 ≤ n ≤ 8. In E 9 , ﬁx a lattice Λ8 in the plane H = x ∈ E 9 : x9 = 0 and take Λ9 = {zu + Λ8 : z ∈ Z} such that S9 + Λ9 is the densest lattice packing of this type. Then k∗ (S9 ) can be realized in S9 + Λ9 . At present we do not know the exact values of δ ∗ (Sn ) for any n ≥ 9, nor the exact values of k∗ (Sn ) for any n ≥ 10, except 24, which will be discussed in Chapter 9.

2.7. Three Mathematical Geniuses: Zolotarev, Minkowski, and Voronoi In order to study a subject that has developed slowly over the centuries it is important to know something of its history and its major contributors.

2.7. Three Mathematical Geniuses

43

Many great mathematicians, such as Blichfeldt, Delone, Dirichlet, Gauss, Hermite, Korkin, Lagrange, Minkowski, Voronoi, Watson, and Zolotarev, have made important contributions to our knowledge of lattice sphere packings or, alternatively, positive deﬁnite quadratic forms. In this section we will brieﬂy introduce three of them, Zolotarev, Minkowski, and Voronoi. For more details about them we refer to Ozhigova [1]. Egor Ivanovich Zolotarev was born in St. Petersburg in 1847, the son of a watch store owner. At the age of seventeen he entered the MathematicalPhysical Faculty of St. Petersburg University, where he attended the lectures of P.L. Chebyshev and A.N. Korkin, with whom he later collaborated on a number of important papers. In 1867 he completed his undergraduate studies and earned the degree of candidate. In 1874 he obtained a doctorate, also from St. Petersburg University, with a dissertation The Theory of Whole Complex Numbers with Application to the Integral Calculus. This remarkable work made him one of the most prominent number theorists of his time. During the summers of 1872 and 1876 the university sent Zolotarev abroad for four months. This gave him the oppourtunity to visit Berlin, Heidelberg, and Paris, to attend lectures of Kirchhoﬀ, Kummer, and Weierstrass, and to meet Hermite. In 1876, Zolotarev was promoted to professor at St. Petersburg University. In the very same year he was elected a member of the St. Petersburg Academy of Sciences. In the summer of 1878, while at his most creative and productive, Zolotarev’s life came to an abrupt and tragic end. On July 2, while preparing to visit relatives in their summer home, he was knocked down by a vehicle and died of blood poisoning on July 19. Besides his joint work with Korkin on positive deﬁnite quadratic forms, some of which was discussed in the previous sections, Zolotarev made important contributions to the arithmetic of algebraic numbers, the theory of algebraic functions, and the theory of best approximation of functions by polynomials. His ideas and methods have had a fundamental inﬂuence on the further development of these subjects. Hermann Minkowski, one of the most remarkable geniuses in the history of mathematics, was born in the town of Aleksoty in Russia (now in Lithuania) in 1864. While he was still a child his family moved to K¨ onigsberg. At the age of ﬁfteen he graduated from the local gymnasium and then studied at the Universities of K¨ onigsberg and Berlin, attending the lectures of Helmholtz, Hurwitz, Kirchhoﬀ, Kronecker, Kummer, Lindemann, Weber, Weierstrass, and others. In 1882, while he was still a student, Minkowski took part in a Paris Academy of Sciences competition on the problem of representing an integer as a sum of ﬁve squares. His paper made a profound impression on the jury, which included Bertrand, Bonnet, Bouquet,

44

2. Quadratic Forms, Lattice Sphere Packings

Hermite, and Jordan. The judges were so impressed that the author was forgiven for writing his paper in German, which was against the rules of the Academy. As a result, in 1883, Minkowski and Smith, a well-known number theorist at Oxford University, shared the Grand Prix des Sciences Math´ematiques of the Paris Academy of Sciences. In 1885, H. Minkowski obtained a doctorate from the University of K¨ onigsberg with a dissertation on positive deﬁnite quadratic forms in n variables. He then began teaching at K¨ onigsberg, and in 1887 he moved to Bonn. In 1892 he became an associate, and in 1894 a full professor. In 1895 he was promoted to the chair of Hilbert, a student colleague and friend of his at K¨ onigsberg. He stayed for only two years and then accepted an oﬀer from ETH-Z¨ urich. In 1902, due to the eﬀorts of Hilbert and Klein, a new chair for mathematics and physics was created at the University of G¨ ottingen. Minkowski took the new chair from 1902 until his sudden death in 1909. Minkowski is known as the founder of the geometry of numbers. In order to investigate positive deﬁnite quadratic forms, he tried to represent the problems in geometric terms, and he then discovered the fundamental principles of geometry of numbers. Minkowski also made important contributions to several other subjects, such as convex geometry and even relativity. There are many mathematical terms such as Minkowski’s theorem, the Minkowski-Hlawka theorem, the Brunn-Minkowski inequality, and the Minkowski metric that bear his name in memory of his great contributions. Georgii Feodosevich Voronoi was born in 1868 to the family of a professor of Russian literature at the Lyceum in Nezhinsk. After graduating from the gymnasium in Priluki in 1885, Voronoi entered the MathematicalPhysical Faculty of St. Petersburg University to study mathematics. When he was still a student at the gymnasium he wrote his ﬁrst mathematical article, which was published in 1885 in the Journal of Elementary Mathematics. During his university years, Voronoi studied the properties of Bernoulli numbers and made a discovery that attracted the attention of A. Korkin and A.A. Markov. He obtained his master’s degree in 1894 with a dissertation On Integral Algebraic Numbers Depending on the Root of an Equation of the Third Degree and a doctorate in 1896 with a dissertation On a Generalization of the Algorithm of Continued Fractions, both from St. Petersburg. For these dissertations he was awarded a Bunyakovskii prize in 1896 by the St. Petersburg Academy of Sciences. In 1894, Voronoi became a professor at the University of Warsaw. In 1907, he was elected corresponding member of the St. Petersburg Academy of Sciences. Unfortunately, he died in 1908, at the age of forty. In 1908 and 1909, Voronoi published three long papers (altogether 287 pages) on positive deﬁnite quadratic forms in the Journal f¨ ur die reine und angewandte Mathematik. These are three of the most important papers in

2.7. Three Mathematical Geniuses

45

this subject and have had a profound inﬂuence on its further development in this century, especially on the work of Baranovskii, Barnes, Delone, ˘ Ry˘skov, Stogrin, Tammela, Venkov, Watson, and others. He also made contributions to analytic number theory, for example to the divisor problem and to the theory of the Riemann zeta function.

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3. Lower Bounds for the Packing Densities of Spheres

3.1. The Minkowski-Hlawka Theorem While it is impossible to determine the values of δ∗ (Sn ) for all dimensions, obtaining reasonable asymptotic bounds for them turns out to be both important and interesting. In 1905, by studying positive deﬁnite quadratic forms, Minkowski [6] proved δ ∗ (Sn ) ≥

ζ(n) , 2n−1

∞ where ζ(x) = k=1 1/k x is the Riemann zeta function, and made a general conjecture for bounded star bodies. Forty years later his conjecture was proved by Hlawka [1]. In this section we prove the Minkowski-Hlawka theorem for centrally symmetric convex bodies. Theorem 3.1 (The Minkowski-Hlawka Theorem). Let C be an ndimensional centrally symmetric convex body. Then δ ∗ (C) ≥

ζ(n) . 2n−1

To prove this theorem we introduce two general lemmas. Lemma 3.1 (Davenport and Rogers [1]). Let f (x) be a continuous function vanishing outside a bounded region, and for a real number λ, write % ∞ % ∞ I(λ) = ··· f (x)dx1 · · · dxn−1 , −∞

−∞

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

48

where x = (x1 , . . . , xn−1 , λ). Let Λ∗ be a lattice in the (n − 1)-dimensional subspace xn = 0 and let α be a ﬁxed positive number. Then there exist a point y = (y1 , . . . , yn−1 , α) and a lattice Λ = {zy + Λ∗ : z ∈ Z} such that

1 det(Λ∗ )

f (u) ≤

u∈Λ, un =0

I(kα).

(3.1)

k∈Z\{0}

Proof. Without loss of generality, we may assume that Λ∗ = Zn−1 . Then f (u) = f (z + ky). k∈Z\{0} z∈Λ∗

u∈Λ, un =0

For convenience we write x = (x1 , x2 , . . . , xn−1 , kα) and assume k > 0. Then, averaging the inner sum on the right-hand side with respect to (y1 , y2 , . . . , yn−1 ) and substituting kyi = xi , we have %

%

1

1

··· 0

0

z∈Λ∗ % k

1

=

0

%

%

1

··· 0

%

%

···

= −∞

z∈Λ∗

%

1

0

0 ∞

f (z + x) dx1 · · · dxn−1

···

z ∈Λ∗ 0≤z ≤k−1 i

1

%

k n−1

k

0

1

=

%

···

k n−1

=

f (z + ky) dy1 · · · dyn−1

0

1

f (z + z + x) dx1 · · · dxn−1

z∈Λ∗

f (z + x) dx1 · · · dxn−1

z∈Λ∗ ∞

−∞

f (x)dx1 · · · dxn−1 = I(kα).

Therefore, %

%

1

··· 0

0

1

f (u) dy1 · · · dyn−1 =

u∈Λ, un =0

I(kα).

k∈Z\{0}

Then there is a point y such that (3.1) holds with det(Λ∗ ) = 1, which proves Lemma 3.1. 2 Lemma 3.2 (Hlawka [1]). Let f (x) be a bounded Riemann integrable function vanishing outside a bounded region, and let be a positive number. Then, there exists a lattice Λ with determinant 1 such that % f (u) < f (x)dx + . u∈Λ\{o}

En

3.1. The Minkowski-Hlawka Theorem

49

Proof. Without loss of generality, we may assume that f (x) is continuous. Let α and β be positive numbers such that β = α1/(1−n) , and denote by Λ∗ the lattice βZn−1 , in the subspace xn = 0. Then det(Λ∗ ) = α−1 .

(3.2)

Since f (x) = 0, it follows that if x is outside a bounded region, f (u) = 0

(3.3)

holds for every point u ∈ Λ∗ \ {o} when α is suﬃciently small. Also, by the deﬁnition of I(λ) in the previous lemma it follows that % lim αI(kα) = f (x)dx. (3.4) α→0

En

k∈Z\{0}

Thus, by (3.2), (3.3), (3.4), and Lemma 3.1, for any > 0 there exist a suitable number α and a suitable point y = (y1 , . . . , yn−1 , α) such that the corresponding lattice, Λ = {zy + Λ∗ : z ∈ Z} , satisﬁes both det(Λ) = 1 and f (u) =

f (u) ≤

u∈Λ, un =0

u∈Λ\{o}

%

αI(kα)

k∈Z\{0}

f (x)dx + .

< En

2

Lemma 3.2 is proved. Proof of Theorem 3.1. It is suﬃcient to show that if v(C) < 2ζ(n),

(3.5)

then there is a lattice Λ with determinant 1 such that 12 C + Λ is a packing. In other words, C ∩ Λ = {o}. (3.6) Let χ(x) be the characteristic function of C and deﬁne f (x) =

∞ k=1

µ(k)χ(kx),

(3.7)

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

50

where µ(k) is the M¨ obius function. It is well known that ∞ µ(k) k=1

and

kn

µ(k) =

k|m

1 ζ(n)

(3.8)

1 if m = 1, 0 otherwise.

(3.9)

=

It follows from (3.7), (3.8), and (3.5) that % f (x)dx = En

=

∞ k=1 ∞ k=1

=

% µ(k)

χ(kx)dx En

µ(k) v(C) kn

v(C) < 2. ζ(n)

(3.10)

Furthermore, denoting the set of primitive points of Λ by Λ , it follows from (3.7) and (3.9) that

f (u) =

∞

f (lu)

l=1 u∈Λ

u∈Λ\{o}

= = =

∞ ∞

µ(k)χ(klu)

u∈Λ l=1 k=1 ∞

u∈Λ m=1

k|m

χ(mu)

µ(k)

χ(u).

(3.11)

u∈Λ

Then, by Lemma 3.2, (3.10), and (3.11) there is a suitable lattice Λ with determinant 1 such that % χ(u) = f (u) < f (x)dx < 2. u∈Λ

u∈Λ\{o}

En

This implies (3.6), and hence Theorem 3.1 is proved.

2

Remark 3.1. The Minkowski-Hlawka theorem is one of the most important results in the geometry of numbers. In the past ﬁve decades, diﬀerent proofs and improvements were achieved by Bateman [1], Cassels [1], Davenport and Rogers [1], Lekkerkerker [1], Macbeath and Rogers [1], [2], [3], Mahler

3.2. Siegel’s Mean Value Formula

51

[1], Maly˘ sev [1], Rogers [6], [7], [9], Sanov [1], Schmidt [1], [4], Schneider [1], Siegel [1], Weil [1], and others. However, none were able to improve the asymptotic order of the lower bound.

3.2. Siegel’s Mean Value Formula A positive deﬁnite quadratic form Q(x) = xSx is called properly reduced if sij ≥ 0 for all index pairs j = i + 1 and if Q(z) ≥ sii whenever (zi , zi+1 , . . . , zn ) = 1. As in the L-S-M reduced case, it can be proved that every positive deﬁnite quadratic form is equivalent to a properly reduced one. Let A be the family of n × n matrices A with determinant 1, and let A1 be the subfamily such that Q(x) = xAA x is properly reduced and the trace of A is nonnegative. Then, denote by P and P1 the sets of points (a11 , a12 , . . . , a1n , a21 , a22 , . . . , a2n , . . . , ann ) 2

in E n such that A ∈ A and A ∈ A1 , respectively. It follows from the deﬁnition of the properly reduced form that A ∈ rint(P1 ) if and only if the corresponding form Q(x) = xAA x = xSx satisﬁes both sij > 0 for j = i + 1 and Q(z) > sii for z ∈ Zn such that (zi , zi+1 , . . . , zn ) = 1 and (zi , zi+1 , . . . , zn ) = (±1, 0, . . . , 0). Thus, routine argument yields that P1 is a fundamental domain of P. In other words, let P1 (U ) be the subset of P corresponding to {AU : A ∈ A1 } and let U be the family of all n × n unimodular matrices U with determinant 1. Then P1 (U ) U ∈U

is a tiling in P. For any Jordan measurable subset J of P, let J be the 2 cone in E n with base J and vertex o. Then, in P we deﬁne a measure σ by σ(J ) = v(J ). Let f (x) be a bounded Riemann integrable function vanishing outside a bounded region and, for convenience, write % F = f (x)dx. En

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

52 Also, for any matrix A we write

Φ(A) =

f (zA ).

z∈Zn \{o}

With these deﬁnitions and notations, in 1945 C.L. Siegel proved the following mean value formula. Lemma 3.3 (Siegel [1]). n 1 # ζ(k) σ(P1 ) = n k=2

%

and

P1

Φ(A)dσ = σ(P1 )F.

Proof. Let I be the (n − 1)-dimensional cube {y = (y2 , y3 , . . . , yn ) : 0 ≤ yi < 1}, and let X be a bounded Lebesgue measurable set in En such that v(X) = F . For convenience, we denote the analogues of A, A1 , P, P1 , and σ for the (n − 1) × (n − 1) matrices by A∗ , A∗1 , P ∗ , P1∗ , and σ∗ , respectively. In addition, we assume that x1 = 0 for any point x ∈ X. Let N be the family of the matrices   x1 0 · · · 0   x2   A= .   ..  D xn

1 y2 0 .. .

··· A∗

yn

   , 

0

where x ∈ X, y ∈ I, A∗ ∈ A∗1 , and D is a diagonal matrix with diagonal elements |x1 |−1/(n−1) , . . . , |x1 |−1/(n−1) , |x1 |−1/(n−1) sign x1 , and let G be 2 its corresponding set of points in E n . Clearly, G is a subset of P. Let χ(x) be the characteristic function of X and let X (A) be the characteristic function of G. Then, % σ(G) = X (A)dσ. P

By routine computation it follows that

% n−1 ∗ ∗ σ (P1 ) χ(x)dx n En n−1 ∗ ∗ = σ (P1 )F. (3.12) n Next, let z be a primitive point of Zn and let U(z) be the set of all unimodular matrices with determinant 1 having ﬁrst column z . For any matrix A∈ rint(P1 (U )), σ(G) =

U ∈U

3.2. Siegel’s Mean Value Formula

53

by the deﬁnitions of I and P1∗ and routine argument it follows that if zA ∈ X, then AU ∈ N for exactly one U ∈ U(z); if zA ∈ X, then AU ∈ N for any U ∈ U(z). Therefore, one has X (AU ) = χ(zA ), (3.13) z∈Z

U ∈U

where Z indicates the set of primitive points of Zn . Since P1 is a fundamental domain in P with respect to U and σ is invariant under the transformation A → AU for any ﬁxed U ∈ U, we have % % σ(G) = X (A)dσ = X (AU )dσ. P

P1 U ∈U

Then, subsitituting (3.12) and (3.13) in this formula, we obtain % n−1 ∗ ∗ χ(zA )dσ = σ (P1 )F. n P1 z∈Z

Since χ(kx) is the characteristic function of k1 X, we have %

χ(zA )dσ =

P1 z∈Z \{o} n

∞ % k=1

= and therefore % n P1

χ( zA )dσ =

z∈Zn \{o}

χ(kzA )dσ

P1 z∈Z

n−1 ζ(n)σ∗ (P1∗ )F n n−1 ζ(n)σ∗ (P1∗ )F n

(3.14)

(3.15)

for every > 0. For any lattice Λ with determinant 1 and any point x, card {Λ ∩ (In + x)} ≤ card {Λ ∩ 2In } . Hence, by approximation it follows that χ( zA ) ≤ c −n

z∈Zn \{o}

z∈Zn \{o}

χ (zA ),

where c is a constant depending only on X and n, and χ (x) is the characteristic function of the unit cube In . Clearly, lim n χ( zA ) = v(X) = F. →0

z∈Zn \{o}

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

54

In addition, c χ (zA ) is integrable over P1 , on account of (3.14) with χ replaced by χ . Hence, letting → 0 in (3.15) and applying Lebesgue’s dominated convergence theorem, we have σ(P1 )F =

n−1 ζ(n)σ∗ (P1∗ )F n

and therefore σ(P1 ) =

(3.16)

n 1 # ζ(k), n k=2

which proves the ﬁrst assertion of our lemma. In addition, it follows from (3.14) and (3.16) that % χ(zA )dσ = σ(P1 )F. P1 z∈Z \{o} n

By a suitable approximation process, one ﬁnds that this formula remains true if χ(x) is replaced by an arbitrary integrable function f (x), which proves the second assertion of our lemma. Lemma 3.3 is proved. 2 Remark 3.2. Siegel’s mean value formula is an improvement of Lemma 3.2. So, one can deduce the Minkowski-Hlawka theorem from it. Remark 3.3. Let f (x1 , x2 , . . . , xk ) be a nonnegative Borel measurable function deﬁned in the space Ekn that vanishes outside a bounded region, and for any n × n matrix A write f (z1 A, z2 A, . . . , zk A). Ψ(A) = zi ∈Zn

By studying the integral

% Ψ(A)dσ, P1

Rogers and Schmidt have successively improved the Minkowski-Hlawka theorem to n log 2 + β , δ ∗ (C) ≥ 2n where β is a suitable constant. So far, this is the best known lower bound of this type. Remark 3.4. For the most interesting case C = Sn , the best known lower bound (n − 1)ζ(n) δ ∗ (Sn ) ≥ 2n−1 is due to Ball [2].

3.3. Sphere Coverings

55

3.3. Sphere Coverings and the Coxeter-Few-Rogers Lower Bound for δ(Sn ) Let Y be a discrete set of points in En . We will call Sn + Y a sphere covering of En if En = (Sn + y). y∈Y

In addition, if Y is a lattice, it will be called a lattice sphere covering of E n . Let Y be the family of sets Y such that Sn + Y is a covering of E n , and let L be the family of lattices Λ such that Sn + Λ is a lattice covering of E n . Then we deﬁne θ(Sn ) = lim inf l→∞ Y ∈Y

and θ∗ (Sn ) = lim inf l→∞ Λ∈L

card{lIn ∩ Y }ωn ln card{lIn ∩ Λ}ωn . ln

Usually, θ(Sn ) and θ∗ (Sn ) are called the density of the thinnest sphere coverings of E n and the density of the thinnest lattice sphere coverings of E n , respectively. It is easy to see that as with the lattice packing density, θ∗ (Sn ) = lim inf Λ∈L

ωn . det(Λ)

Let Sn + X be a sphere packing in E n with the maximal density δ(Sn ). Without loss of generality, we assume that for any point y ∈ En , int(Sn + y) ∩ (Sn + X) = ∅. Then, 2Sn + X is a sphere covering in E n . By the deﬁnitions of δ(Sn ) and θ(Sn ), one has card{lIn ∩ X}v(2Sn ) ln l→∞ ≥ θ(2Sn ) = θ(Sn )

2n δ(Sn ) = lim sup

and therefore

θ(Sn ) . (3.17) 2n Besides the interest in θ(Sn ) itself, by (3.17) a lower bound for θ(Sn ) will imply a lower bound for δ(Sn ). So, in this section we introduce both an upper bound and a lower bound for θ(Sn ). δ(Sn ) ≥

Theorem 3.2 (Rogers [8]). When n ≥ 3, θ(Sn ) < n log n + n log log n + 5n.

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

56

Proof. For convenience, in this proof let Sn be the n-dimensional sphere of volume 1. Let l be a large positive number and let m be a large positive integer. Also, write I = {(x1 , x2 , . . . , xn ) : 0 ≤ xi < l} and Λ = lZn . It is easy to see that Sn + Λ is a packing. Let X = {x1 , x2 , . . . , xm } be a subset of I. We will study the system Sn + X + Λ.

(3.18)

Let χ(x) be the characteristic function of Sn . Then the characteristic function of the set E = E n \ (Sn + X + Λ) is

m # χ(x − xi − u) . χ (x) = 1− i=1

u∈Λ

Since this function is periodic in each coordinate with period l, E is a set with asymptotic density equal to v(I ∩ E )/v(I), where % v(I ∩ E ) = χ (x)dx. I

Let M be the mean value of v(I ∩ E ), taken over all distributions of x1 , x2 , . . . , xm throughout the cube I. Then

% % % % 1 M = mn ··· χ (x) dx dx1 dx2 · · · dxm l I I I I

' % &# m % 1 = mn 1− χ(x − xi − u) dxi dx l I i=1 I u∈Λ

' % &# m % 1 = mn ln − χ(−y)dy dx l I i=1 u∈Λ I−x+u

m % 1 = mn ln ln − χ(−y)dy l En

m 1 = ln 1 − n . l Hence, there is a set X such that the asymptotic density of the corresponding E does not exceed (1 − l−n )m . Let η be any number satisfying 0 < η ≤ 1, and let m be the maximal cardinality of the set Y = {y1 , y2 , . . . , ym } such that ηSn + Y + Λ is a packing in E . Then, one has

m 1 m η n ≤ ln 1 − n l

(3.19)

3.3. Sphere Coverings

57

and therefore m ≤

m ln 1 . 1 − ηn ln

(3.20)

Since m is maximal and 0 < η ≤ 1, it follows from the choice of the two systems (3.18) and (3.19) that (1 + η)Sn + X ∪ Y + Λ is a covering of E n . The density of this covering is (1 + η)n (m + m ) . ln Therefore, by (3.20) and the deﬁnition of θ(Sn ), &

m ' m 1 1 . θ(Sn ) ≤ min (1 + η)n n + n 1 − n l, m, η l η l Then, taking m 1 = n log(1/η) and η = n l n log n successively, one has (1 + η)n n log(1/η) + η −n e−n log(1/η) 0<η≤1 min (1 + η)n 1 + n log(1/η) 0<η≤1 min enη 1 + n log(1/η) 0<η≤1 e1/ log n 1 + n log(n log n) 2 1+ (n log n + n log log n + 1) log n n log n + n log log n + 5n.

θ(Sn ) ≤ min = < ≤ < <

2

Theorem 3.2 is proved. For the lattice case, we have the following analogue.

Theorem 3.3 (Rogers [11]). For n ≥ 3, there is a constant c such that θ∗ (Sn ) ≤ c (n log2 n)

log2

√ 2πe

.

Remark 3.5. In fact, Rogers’ original results dealt with not only spheres, but general convex bodies. Let K be an n-dimensional convex body, n ≥ 3. Rogers proved that θ(K) ≤ n log n + n log log n + 5n

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

58 and

θ∗ (K) ≤ nlog2 n+c log2 log2 n ,

where c is a suitable constant. Let T = conv{v0 , v1 , . . . , vn } be a regular simplex with side length 2(n + 1)/n. The n + 1 unit spheres Sn + v0 , Sn + v1 , . . . , Sn + vn just cover T . We deﬁne

n i=0 v((Sn + vi ) ∩ T ) τn = v(T ) (n + 1)v((Sn + v0 ) ∩ T ) = . v(T ) As a counterpart of Theorem 3.2, we have the following result. Theorem 3.4 (Coxeter, Few, and Rogers [1]). θ(Sn ) ≥ τn . To prove this theorem, ﬁrst let us introduce a basic concept and an important lemma. Let P be a polytope in En with vertices v0 , v1 , . . . , vk , and write Vi = {vi + λ(x − vi ) : x ∈ P, λ ≥ 0} . Then we call φ(vi ) = s(bd(Sn + vi ) ∩ Vi ) the solid angle of P at vi . Lemma 3.4 (Coxeter, Few, and Rogers [1]). Let T be an ndimensional simplex, with vertices v0 , v1 , . . . , vn , contained in Sn . Then, 1 φ(vi ) ≥ nτn , v(T ) i=0 n

where equality holds when T is a regular simplex inscribed in Sn . Proof. Let vi = (vi1 , vi2 , . . . , vin ) and suppose temporarily that v0 = o. We introduce new coordinates (y1 , y2 , . . . , yn ) related to the original coordinates (x1 , x2 , . . . , xn ) by the equations xi = v1i y1 + v2i y2 + · · · + vni yn ,

i = 1, 2, . . . , n.

Then the new coordinates of v0 , v1 , . . . , vn are (0, 0, . . . , 0), (1, 0, . . . , 0), (0, 0, . . . , 1), respectively, and v(T ) =

ν , n!

(3.21)

3.3. Sphere Coverings

59

where ν is the absolute value of the determinant of (vij ). Since % nωn Γ(n/2) , e− x,x dx = 2 n E it follows that

(% e− x,x dx e− x,x dx yi ≥0 En % ∞ % ∞% ∞ 2ν ··· e− x,x dy1 dy2 · · · dyn = nωn Γ(n/2) 0 0 0 % ∞% ∞ % ∞ 2ν = ··· e−Q(y,v0 ) dy1 dy2 · · · dyn , nωn Γ(n/2) 0 0 0

φ(v0 ) = nωn

%

where Q(y, v0 ) =

n

vj − v0 , vk − v0 yj yk .

j, k=1

In the general case, keeping (3.21) in mind, we have % ∞ % ∞% ∞ 2n! φ(vi ) ··· e−Q(y,ϕi ) dy1 dy2 · · · dyn , = v(T ) Γ(n/2) 0 0 0 where Q(y, ϕi ) =

n

(3.22)

vϕi (j) − vϕi (0) , vϕi (k) − vϕi (0) yj yk

j, k=1

and ϕi is any one of the n! permutations of the integers 0, 1, . . . , n with ϕi (0) = i. Let ϕ be an arbitrary permutation of the integers 0, 1, . . . , n. It follows from (3.22) and the deﬁnition of ωn that

% ∞ % ∞% ∞ n! φ(vi ) 2n! = ··· e−Q(y,ϕ) dy1 dy2 · · · dyn . v(T ) Γ(n/2) 0 0 0 ϕ Then, by the arithmetic mean and geometric mean inequality, we have

1

% % % ∞ (n+1)! Q(y,ϕ) φ(vi ) 2(n + 1)! ∞ ∞ ϕ ≥ ··· e dy1 dy2 · · · dyn , v(T ) Γ(n/2) 0 0 0 (3.23) where equality holds when T is regular. Writing n 1 s2 = vj − vk , vj − vk , (3.24) n(n + 1) j, k=0

since vi − vj , vi − vj = vi − vk , vi − vj + vk − vj , vi − vj ,

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

60 routine computation yields

1 vϕ(j) − vϕ(0) , vϕ(k) − vϕ(0) = (n + 1)! ϕ

)

1 2 2s 2

s

if j = k, if j = k.

(3.25)

Then, it follows from (3.23) and (3.25), with the substitutions syi = xi , that

% % % ∞ φ(vi ) 2(n + 1)! ∞ ∞ · · · e−Q(x) dx1 dx2 · · · dxn , (3.26) ≥ v(T ) Γ(n/2)sn 0 0 0 where Q(x) =

xj xk ,

j≤k

and equality holds when T is regular. Since the integral in (3.26) is a constant, by checking a special simplex, (3.26) yields

n/2 φ(vi ) 2(n + 1) ≥n τn . v(T ) ns2

(3.27)

Since T ⊆ Sn , it follows from (3.24) that

n n n 2 s = (n + 1) vi , vi − vi , vi n(n + 1) i=0 i=0 i=0 2

≤

2(n + 1) , n

(3.28)

where equality holds if T is a regular simplex inscribed in Sn . By (3.27) and (3.28), Lemma 3.4 is proved. 2 Proof of Theorem 3.4. For any positive number , let Sn + X be a covering of E n such that lim inf l→∞

card{lIn ∩ X}v(Sn ) ≤ θ(Sn ) + . ln

(3.29)

Without loss of generality, we assume that card {X ∩ bd(λSn + y)} ≤ n + 1

(3.30)

for any positive number λ ≤ 1 and any point y ∈ En . Let D(x) be the Dirichlet-Voronoi cell associated with X at x ∈ X, and let V be the set of vertices of these Dirichlet-Voronoi cells. For every v ∈ V , by (3.30) there are exactly n + 1 points x0 , x1 , . . . , xn of X such that v is a vertex of D(xi ). Denote the simplex with vertices x0 , x1 , . . . , xn by T (v). The family of simplices T (v), v ∈ V , tile the space En . These

3.3. Sphere Coverings

61

simplices are Delone simplices. Together they form a Delone triangulation of the space E n . Clearly, it follows from (3.30) that the diameters of the Delone simplices associated with X are bounded by 2 from above. Let l be a suﬃciently large positive number. By (3.29) and Lemma 3.4 it follows that θ(Sn ) + 2 ≥ ≥

card{lIn ∩ X}s(Sn ) nln

n i=0 φ(xi ) T (v)∈lIn

≥

nln T (v)∈lIn v(T (v))τn

ln v((l − 2)In ) ≥ τn ln and therefore θ(Sn ) ≥ τn . 2

Thereom 3.4 is proved.

As a direct consequence of (3.17), Theorem 3.4, and Corollary 7.2, we have the following lower bound for δ(Sn ). Theorem 3.5 (Coxeter, Few, and Rogers [1]). δ(Sn ) ≥

τn n ∼ n √ . n 2 2 e e

Remark 3.6. This lower bound is weaker than both Schmidt’s lower bound mentioned in Remark 3.3 and Ball’s lower bound mentioned in Remark 3.4. However, its proof shows a connection between sphere packings and sphere coverings. Remark 3.7. As a counterpart of sphere packing, sphere covering itself is also a fascinating research subject. Many authors have made contributions to the study of θ∗ (Sn ). The known exact results are summarized in the following table. n

θ∗ (Sn )

Author

2

2π √ 3 3 √ 5 5π 24

Kershner [1]

3 4 5

2

2π √ 5 5 √ 2 245 35π √ 3888 3

Bambah [1], Barnes [2], and Few [2] Delone and Ry˘skov [1] Ry˘skov and Baranovskii [1]

3. Lower Bounds for δ ∗ (Sn ) and δ(Sn )

62

As for the exact value of θ(Sn ), our only knowledge is 2π θ(S2 ) = θ∗ (S2 ) = √ , 3 3 which was discovered by Kershner [1].

3.4. Edmund Hlawka Edmund Hlawka was born in 1916 in Bruck, Austria. Inﬂuenced by his schoolteachers, he became interested in mathematics and physics. In 1934 he became a student at the University of Vienna. At ﬁrst he studied both mathematics and physics, but soon devoted his time solely to mathematics. In 1938 he obtained a doctorate under the supervision of W. Wirtinger. Impressed by his dissertation, C.L. Siegel invited him to be an assistant in G¨ ottingen. However, Hlawka preferred to stay in Vienna. In 1948, at the age of 32, he became a professor of mathematics at the University of Vienna, where he worked for 33 years. In 1981 he moved to the Technical University of Vienna, from which he retired in 1987. He was also a guest of many well-known institutions including the Institute for Advanced Study, the University of Paris at Sorbonne, California Institute of Technology, ETH-Z¨ urich, and the University of Freiburg. Professor Hlawka has made fundamental contributions to the geometry of numbers. In particular, the Minkowski-Hlawka theorem is a classic result in mathematics. It has been improved and generalized by many masters such as Cassels, Davenport, Mahler, Maly˘sev, Rogers, Schmidt, Schneider, Siegel, and Weil. However, so far, no result is essentially better than his original, as regards the exponent. He has also made inﬂuential contributions to the theory of uniform distributions, for example, in discrepancy theory. In 1990, Springer-Verlag published his Selecta which contains his most important mathematical works. For his distinguished contributions to mathematics, Professor Hlawka ¨ has received many honors. He is a member of both the Osterreichische Akademie der Wissenschaften and the Deutsche Akademie der Naturforscher “Leopoldina,” and is a corresponding member of the Reinische-Westf¨ alische Akademie der Wissenschaften, the Bayerische Akademie der Wissenschaften, and the Bologna Academy of Sciences. He has been confered honorary doctorates by Universities of Vienna, Erlangen, Salzburg, and Graz, and the Technical University of Graz. He has also been the recipient of many prizes. ¨ For example, the Erwin-Schr¨ odinger prize of the Osterreichische Akademie der Wissenschaften, the Gauss-medal of the Akademie der Wissenschaften

3.4. Edmund Hlawka

63

Berlin, and the Dannie-Heinemann prize of the G¨ ottinger Akademie der Wissenschaften.

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4. Lower Bounds for the Blocking Numbers and the Kissing Numbers of Spheres

4.1. The Blocking Numbers of S3 and S4 The blocking number b(K) of a convex body K is the smallest number of nonoverlapping translates K + xi , all of which touch K at its boundary, that can prevent any other translate K + x from touching K. Although it is as natural as the kissing number, the blocking number was introduced by Zong [2] only in 1994. Clearly, the problem of determing the blocking number of a convex body itself is both interesting and challenging. However, so far our knowledge of the blocking number is very limited. For a general convex body K it follows from the deﬁnition that b(K) ≤ k(K).

(4.1)

As for k(K), one can easily deduce that b(K) = b(D(K)), where D(K) is the diﬀerence body of K. Therefore, since D(K) is centrally symmetric, we can deal solely with the centrally symmetric case when we study the blocking numbers of convex bodies. In this way, by applying (1.28) it can be deduced that b(K) = 4 for every two-dimensional convex domain.

66

4. Lower Bounds for b(Sn ) and k(Sn )

Clearly, like the kissing number, spheres are the most interesting case for the blocking numbers. In this section we will determine the blocking numbers of S3 and S4 . Theorem 4.1 (Dalla, Larman, Mani-Levitska, and Zong [1]). b(S3 ) = 6. Proof. By routine argument it can be veriﬁed that the six nonoverlapping unit spheres centered at ±(2, 0, 0), ±(0, 2, 0), and ±(0, 0, 2) prevent any other unit sphere from touching the one centered at o. Thus, b(S3 ) ≤ 6.

(4.2)

. −x1 Ω2

Ω1 Figure 4.1 On the other hand, suppose that X = x1 , x2 , . . . , xb(S3 ) is a set such that every S3 + xi touches S3 at its boundary and S3 + X prevents any other sphere S3 + x from touching S3 , and write Ωi = bd(2S3 ) ∩ (int(2S3 ) + xi ). Then

b(S3 )

bd(2S3 ) =

Ωi .

i=1

Clearly, we have

u1 ou2 < 2π/3

whenever both u1 and u2 belong to one of the b(S3 ) caps Ωi . Thus, any great circle of bd(2S3 ) intersects at least four of the caps. If, without loss of generality, −x1 ∈ Ω2 , then Ω1 ∩ Ω2 = ∅.

4.1. The Blocking Numbers of S3 and S4

67

So, bd(2S3 ) has a great circle that intersects neither Ω1 nor Ω2 (see Figure 4.1). Thus, we have b(S3 ) ≥ 6. (4.3) 2

Our theorem follows from (4.2) and (4.3). Theorem 4.2 (Dalla, Larman, Mani-Levitska, and Zong [1]). b(S4 ) = 9. To prove this theorem we need two general results.

Lemma 4.1. Let T be a simplex inscribed in Sn . Then T ∩ (rSn + x) has maximal volume for every r if and only if T is regular and x = o. Proof. Clearly, the lemma is true when n = 1. Assume it is true when n = m − 1. We proceed to prove it for n = m. v0 v0

V

V

H ∩ Sm

Figure 4.2 Suppose that T = conv{v0 , v1 , . . . , vm } is an extreme simplex in E m but the facet conv{v1 , v2 , . . . , vm } is irregular. Let H be the hyperplane determined by {v1 , v2 , . . . , vm }, let v0 be the point of Sm with maximal distance from H, let conv{v1 , v2 , . . . , vm } be a regular simplex inscribed in H ∩ Sm , and write T = conv{v0 , v1 , . . . , vm }. Also, denote the cones with vertices v0 and v0 over H ∩ Sm by V and V , respectively. Clearly, T ∩ (H + tv0 ) is a simplex inscribed in the (m − 1)-dimensional sphere V ∩ (H + tv0 ), T ∩ (H + tv0 ) is a regular simplex inscribed in the (m − 1)dimensional sphere V ∩ (H + tv0 ) (see Figure 4.2), and the second sphere is not smaller than the ﬁrst one. Then, we have v(T ∩ (rSm + x)) % ∞ = s([T ∩ (H + tv0 )] ∩ [(rSm + x) ∩ (H + tv0 )])dt 0

68

4. Lower Bounds for b(Sn ) and k(Sn ) % <

∞

s([T ∩ (H + tv0 )] ∩ [(rSm + x ) ∩ (H + tv0 )])dt

0

= v(T ∩ (rSm + x )) for a suitable r, where x is a point on the line determined by o and v0 such that xx is perpendicular to v0 . Lemma 4.1 follows. 2 Lemma 4.2. Let T be a spherical simplex lying within a cap Ω with radius ρ of the unit sphere Sn . Then T has maximal (n − 1)-dimensional measure if and only if it is regular. Proof. Let T be the simplex with the same vertices as T , and let V be the cone with vertex o over T . The simplex is inscribed in an (n − 1)dimensional sphere with center u and radius ρ. Then, we have % 2 s(T ) = c e− x dx, V

%

where

e− x dx. 2

c = nωn / En

Regarding V as a union of pieces λT , 0 ≤ λ < ∞, we have

% ∞ % − x 2 e dλT dλ s(T ) = c 0 λT %

% ∞ n−1 −λ2 x =c λ e dT dλ. 0

(4.4)

T

In the simplex T , writing x = u + y, we have % % ρ 2 2 −λ2 x 2 −λ2 u e dT = e e−λ r µ(r)dr,

(4.5)

0

T

where µ(r) indicates the (n − 2)-dimensional measure of T ∩ (bd(rSn ) + u). Abbreviating s(T ∩ (rSn + u)) to s(r), it is clear that µ(r) =

ds(r) . dr

Then, by Lemma 4.1, % ρ *ρ % ρ 2 2 * −λ2 r 2 −λ2 r 2 e µ(r)dr = e s(r)* + 2λ2 re−λ r s(r)dr 0 0 % ρ0 2 2 2 2 = e−λ r s(ρ) + 2λ2 re−λ r s(r)dr %0 ρ 2 2 −λ2 r 2 ≤e s (ρ) + 2λ2 re−λ r s (r)dr, 0

(4.6)

4.1. The Blocking Numbers of S3 and S4

69

where s indicates the corresponding value in the regular case, and equality holds if and only if T is regular. Lemma 4.2 follows from (4.4), (4.5), and (4.6). 2 Proof of Theorem 4.2. Write e1 = (1, 0, 0, 0), e2 = (0, 1, 0, 0), e3 = (0, 0, 1, 0), e4 = (0, 0, 0, 1), e = (1, 1, 1, 1), ui = √

2 (ei − e), 1 + 4 2 − 2

and

1 (−2ei − e). 1 + + 2 It can be veriﬁed that when is small, S4 + {e, u1 , u2 , u3 , u4 , v1 , v2 , v3 , v4 } blocks any other translate of S4 from touching it. Thus, vi = √

b(S4 ) ≤ 9.

(4.7)

By considering cap coverings, one can deduce b(S4 ) > 8 from the following statement “Let P be a four-dimensional polytope with eight facets such that S4 ⊂ P. Then max x ≥ 2, x∈P

where equality holds if and only if P is a cube with side length 2.” Let P be the dual of P , P = {x : x, y ≤ 1, y ∈ P }. This statement is equivalent to the following assertion. Assertion 4.1. Let P be a polytope inscribed in S4 with eight vertices. Then α(P ) =

min

x∈bd(P )

x ≤ 12 ,

where equality holds if and only if P is a regular cross polytope I . Now we proceed to prove this assertion. It is obvious that α(I ) = and, for any four-dimensional polytope P such that S4 ⊂ P ,

1 2

α(P )S4 ⊂ P . As usual, we use fi to indicate the number of the i-dimensional faces of P . In addition, without loss of generality, we may assume that P is simplicial. In other words, all its facets are simplices. We study two cases. Case 1. α(P ) ≥ 12 and f3 ≤ 16. Projecting the facets of P from o to the surface of S4 , we get f3 spherical simplices, say T1 , T2 , . . ., Tf3 . Let s be the surface area of the corresponding spherical simplex induced by I . It follows from Lemma 4.2 that s(Ti ) ≤ s ,

70

4. Lower Bounds for b(Sn ) and k(Sn )

where equality holds if and only if Ti is induced by a facet of I . On the other hand, we have f3 s(Ti ) = s(S4 ). i=1

Thus, P must be a regular cross polytope. Case 2. α(P ) ≥ 12 and f3 > 16. By the Dehn-Sommerville equations we have f0 = −f0 + 2f1 − 3f2 + 4f3 , f1 = f1 − 3f2 + 6f3 , and hence f1 = f0 + f3 ≥ 25.

Then, since P has 8 vertices, one vertex of P is connected to every other by edges. This means that P has a facet F that intersects all the other facets. Let x be the touching point of F on bd(S4 ), let p(x) be the point of bd(P ) in the opposite direction to x, and assume that max o, y ≤ 2. y∈F

Then simple computation yields that max x ≥ o, p(x) > 2, x∈P

which contradicts α(P ) ≥ 12 . As a conclusion, Assertion 4.1 is proved. Thus, we have b(S4 ) > 8. Then Theorem 4.2 follows from (4.7) and (4.8).

(4.8) 2

To end this section, we propose two conjectures. Conjecture 4.1. Let P be an n-dimensional polytope with 2n facets and Sn ⊂ P . Then √ max x ≥ n, x∈P

where equality holds if and only if P is a cube of side length 2. This conjecture is open for n ≥ 5. Conjecture 4.2 (Zong [2]). 2n ≤ b(K) ≤ 2n . Examples show that both 2n and 2n can be realized by b(K) for certain n-dimensional convex bodies. More surprisingly, they can be realized by convex bodies arbitrarily near to each other, one of which being the cube.

4.2. The Shannon-Wyner Lower Bound for b(Sn ) and k(Sn )

71

4.2. The Shannon-Wyner Lower Bound for Both b(Sn ) and k(Sn ) By studying the blocking numbers of spheres, we can deduce the following famous lower bounds. Theorem 4.3 (Shannon [1] and Wyner [1]). k(Sn ) ≥ b(Sn ) ≥ 20.2075...n(1+o(1)) . Proof. Let 12 Sn + xi , i = 1, 2, . . . , b(Sn ), be b(Sn ) nonoverlapping spheres, all of which touch 12 Sn at its boundary, and that prevent any other sphere 1 1 2 Sn + x from touching 2 Sn . Considering the corresponding system Sn , Sn + x1 , . . . , Sn + xb(Sn ) , we have

b(Sn )

bd(Sn ) ⊂

(int(Sn ) + xi ) .

i=1

In other words, all the caps Ωi = bd(Sn ) ∩ (Sn + xi ) together form a cap covering of bd(Sn ). Thus,

b(Sn )

s(Sn ) ≤

s(Ωi ).

(4.9)

i=1

It is well-known that s(Sn ) = nωn =

nπ n/2 . Γ(n/2 + 1)

(4.10)

On the other hand, s(Ωi ) = ≤

(n − 1)π(n−1)/2 Γ((n + 1)/2) (n − 1)π(n−1)/2 2Γ((n + 1)/2)

%

1

(1 − x2 )(n−3)/2 dx 1/2

(n−3)/2 3 . 4

(4.11)

Then, by (4.9), (4.10), and (4.11), it follows that (n−3)/2 nπn/2 (n − 1)π(n−1)/2 3 ≤ b(Sn ) Γ(n/2 + 1) 2Γ((n + 1)/2) 4 and therefore b(Sn ) ≥

(n−3)/2 1 4 ≥ 20.2075...n(1+o(1)) . n 3

(4.12)

72

4. Lower Bounds for b(Sn ) and k(Sn )

By (4.3) and (4.12), Theorem 4.3 is proved.

2

Remark 4.1. So far, the Shannon-Wyner lower bound is the best known for k(Sn ). However, examining the proof of Theorem 4.3, it is reasonable to conjecture that the exact value of k(Sn ) is much larger than 20.2075n when n is large. Remark 4.2. Despite the similarities in their deﬁnitions, there is no close relationship between the kissing number and the blocking number except (4.1). Examples of Zong [2] show that when n is suﬃciently large, there are two n-dimensional convex bodies K1 and K2 in E n such that b(K1 ) < b(K2 )

and k(K1 ) > k(K2 ).

4.3. A Theorem of Swinnerton-Dyer Despite the regularity of lattices, our knowledge of the asymptotic behavior of k ∗ (Sn ), when n is large, is very limited. In this section we discuss a simple result concerning k∗ (K) and the densest lattice packings of a general convex body K in E n . For this purpose we denote by k(K, Λ) the number of translates K + u, u ∈ Λ, that touch K at its boundary. Theorem 4.4 (Swinnerton-Dyer [1]). If K + Λ is one of the densest lattice packings of K, then k(K, Λ) ≥ n(n + 1). Proof. Clearly, K + Λ is a packing if and only if 12 D(K) + Λ is a packing, and k(K, Λ) = k( 12 D(K), Λ). Without loss of generality, we may assume that K is centrally symmetric and centered at o. Further we may assume that Λ = Zn . Let ±u1 , ±u2 , . . . , ±um be the points of Zn such that K + uk , k = 1, 2, . . . , m, touch K at its boundary. Then, uk = (u1k , u2k , . . . , unk ) ∈ bd(2K) for k = 1, 2, . . . , m. Let I be the n × n unit matrix. Suppose that m < n(n + 1)/2. We proceed to show that for suitable choices of an n × n matrix A and a small number η, Λ = (I + ηA)Zn is a packing lattice of K and det(Λ ) < 1 = det(Λ).

4.3. A Theorem of Swinnerton-Dyer

73

Let vk = (v1k , v2k , . . . , vnk ) be a unit exterior norm of 2K at uk . Since m + 12 n(n + 1) < n2 , there is a nonzero symmetric matrix A with elements aij such that uk A, vk = aij ujk vik = 0 i, j

for k = 1, 2, . . . , m. Writing wk = uk (I + ηA), it is easy to see that wk ∈ int(2K),

k = 1, 2, . . . , m.

Thus, K + Λ is a packing of K. To calculate det(Λ ), we have det(Λ ) = 1 + c1 η + c2 η 2 + · · · + cn η n , where c1 =

n

aii ,

c2 =

i=1

aii ajj − a2ij , i<j

and 2c2 − c21 = −2

i<j

a2ij −

n

a2ii < 0.

i=1

Thus, we can choose a small η with proper sign such that det(Λ ) < 1 = det(Λ). By this contradiction Theorem 4.4 is proved.

2

Remark 4.3. As a counterpart of Swinnerton-Dyer’s theorem, Gruber [1] proved that in the sense of Baire category most n-dimensional convex bodies K satisfy k(K, Λ) ≤ 2n2 for all of their densest packing lattices. Clearly, by Theorem 2.6 and Remark 2.5 both S7 and S8 do not belong to this category. Remark 4.4. As a consequence of Swinnerton-Dyer’s theorem, we have k ∗ (Sn ) ≥ n(n + 1). This lower bound is too small, even when n = 4. Inductively, it can be deduced that k∗ (Sn ) ≥ k∗ (Sn−1 ) + 2n.

74

4. Lower Bounds for b(Sn ) and k(Sn )

For more lower bounds for k∗ (Sn ) and k(Sn ), especially the constructive ones, we refer to the next chapter.

4.4. A Lower Bound for the Translative Kissing Numbers of Superspheres Let α be a number satisfying α ≥ 1. The set

1/α n n α Sn (α) = x ∈ E : |xi | ≤1 i=1

is called an n-dimensional supersphere. By Minkowski’s inequality,

1/α

1/α

1/α n n n |θxi + (1 − θ)yi |α ≤θ |xi |α + (1 − θ) |yi |α i=1

i=1

i=1

whenever 0 ≤ θ ≤ 1. It follows that Sn (α) is a centrally symmetric convex body. Also, it can be veriﬁed that Sn (1) is a cross polytope, Sn (2) is a sphere, and Sn (∞) is a unit cube. In this section we prove the following result. Theorem 4.5 (Larman and Zong [1]). 30.1072...n(1+o(1)) ≤ k(Sn (α)) ≤ 3n . To prove this theorem, ﬁrst we introduce a basic concept and a simple lemma. Let C be an n-dimensional centrally symmetric convex body centered at o. Then, the Minkowski metric given by C is deﬁned by ) 0, if x = y, x, yC =

x−y , otherwise,

p(x−y) where p(x) denotes the boundary point of C in the direction of x, and x denotes the Euclidean norm of x. Lemma 4.3 (Zong [5]). The translative kissing number k(C) of C is the maximal number of points xi ∈ bd(C) such that xi , xj C ≥ 1,

i = j.

Proof. Suppose X = {x1 , x2 , . . . , xk } is a subset of bd(C) such that xi , xj C ≥ 1

4.4. A Lower Bound for k(Sn (α))

75

for every pair of distinct points xi and xj . Then, by symmetry it follows that C touches each of the k translates C + 2xi , xi ∈ X, at its boundary, and (int(C) + 2xi ) ∩ (int(C) + 2xj ) = ∅ whenever xi = xj . Therefore, k(C) ≥ card{X}. On the other hand, if X is a set of points such that C + {o} ∪ {2X} is an optimal kissing conﬁguration of C, then X ⊂ bd(C), and x, yC ≥ 1 holds for every pair of distinct points x and y of X. Thus, Lemma 4.3 follows. 2 Proof of Theorem 4.5. It is easy to see that x, ySn (α) =

n

1/α |xk − yk |

α

.

(4.13)

k=1

In order to prove Theorem 4.5, ﬁrst let us introduce a basic combinatorial result. Assertion 4.2. Let m be a positive integer with m ≤ n. Also let Z∗ be a n set of integer points zi = (zi1 , zi2 , . . . , zin ) with |zik | ≤ 1, k=1 |zik | = m, and n |zik − zjk | ≥ m (4.14) k=1

whenever i = j. Then, k(Sn (α)) ≥ card{Z ∗ }. Taking

Y = yi = m−1/α zi : zi ∈ Z ∗ ,

we have n k=1

1/α |yik |

α

1/α n 1 α = |zik | m k=1

1/α n 1 = |zik | = 1, m k=1

76

4. Lower Bounds for b(Sn ) and k(Sn )

which means that yi ∈ bd(Sn (α)). On the other hand, by (4.13) and (4.14) it follows that

1/α n 1 yi , yj Sn (α) = |zik − zjk |α m k=1

1/α n 1 ≥ |zik − zjk | ≥1 m k=1

whenever i = j. Hence, by Lemma 4.3, k(Sn (α)) ≥ card{Y } = card{Z ∗ }, which proves Assertion 4.2. Writing

n Z = z ∈ Zn : |zi | ≤ 1, |zi | = m ,

i=1

a simple combinatorial argument yields that

n m card{Z } = 2 . m

(4.15)

Similarly, writing h(m) = m/2 + 1, for any point z ∈ Z there are at most

m n − h(m) m−h(m) g(n, m) = 2 (4.16) h(m) m − h(m) points w ∈ Z such that n

|zi − wi | < m.

i=1

Deﬁning

f (n, m) = max {card{Z ∗ }} ,

by (4.15) and (4.16) it follows that

n m 2 − f (n, m)g(n, m) ≤ 0 m and therefore n

2m m g(n, m)

−1

−1 n m n − h(m) = 2h(m) . m h(m) m − h(m)

f (n, m) ≥

(4.17)

4.4. A Lower Bound for k(Sn (α))

77

Writing l = n/m, by Stirling’s formula and detailed computation we have n(1+o(1)) f (n, n/l) ≥ 21−3/2l (2l − 1)1/2l−1 l .

(4.18)

It can be shown by routine computation that the function f (x) = 21−3/2x (2x − 1)1/2x−1x attains its maximum (4.18) it follows that

9 8

at x = 92 . Therefore, by Assertion 4.2, (4.17), and

k(Sn (α)) ≥ f (n, 2n/9) ≥

n(1+o(1)) 9 8

= 30.1072...n(1+o(1)) . Theorem 4.5 follows from this together with Theorem 1.1.

2

Remark 4.5. In 1985 Milman [1] proved the following theorem: Let > 0 and let C be an n-dimensional centrally symmetric convex body. Then there are subspaces El ⊂ E m of E n with l ≥ c( )n and an ellipsoid E in El such that E ⊆ p(C ∩ E m ) ⊆ (1 + )E, where p(x) indicates the orthogonal projection from En onto E l . As a corollary to this result, Talata [1] deduced that there exists a positive constant c such that k(K) ≥ 3cn holds for every n-dimensional convex body K. In fact, Bourgain knew this result much earlier, but he did not publish it.

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5. Sphere Packings Constructed from Codes

5.1. Codes Let F be a ﬁnite ﬁeld. A code ℵ over F of length n is simply a subset of F n . A typical point, or codeword, of ℵ has the form u = (u1 , u2 , . . . , un ), where ui ∈ F . For convenience, we say that a codeword or point is of type [µk |ν l | · · ·] if µ = |ui | for k choices of ui , ν = |ui | for l choices of ui , etc. The Hamming distance between two codewords u and v of ℵ is the number of coordinates at which they diﬀer, and is denoted by u, vH . The weight of a codeword u, w(u), is the number of its nonzero coordinates. Thus, u, vH = w(u − v). The signal transmission process can be described as encoder m

6

m is mapped into u

channel u

-

error v is added

decoder u +v

message

most likely v can be eliminated

m6 estimate

Thus, in this process the number of errors that can be corrected by ℵ is determined by its minimum Hamming distance d(ℵ) = min u, vH . u,v∈ℵ u=v

80

5. Sphere Packings Constructed from Codes

In coding theory, one of the main problems is to determine m(F, n, d), the maximum number of codewords in any code over F of length n and minimal Hamming distance d. When F = {0, 1}, determining m(F, n, d) is also an interesting geometric problem. In fact, in this case m(F, n, d) is the maximal number of vertices of the unit cube In with √ the property that any two of them are at Euclidean distance at least d apart. Usually, a code of length n containing m codewords and with minimal Hamming distance d is said to be an (n, m, d) code. For convenience, we assume that every code contains the zero codeword. A code ℵ over F is called linear if µu + νv ∈ ℵ holds whenever u, v ∈ ℵ and µ, ν ∈ F. In other words, a linear code is an F -module. Then, it is well-known from algebra that ℵ can be spanned by the rows of a generator matrix A = (IB), where I is the k × k unit matrix, and card{ℵ} = q k , where q = card{F }. For convenience, a k-dimensional linear code of length n with minimal Hamming distance d is said to be an [n, k, d] code (or sometimes an [n, k] code). A code ℵ over F is called cyclic if whenever (u0 , u1 , . . . , un−1 ) is a codeword so is (un−1 , u0 , u1 , . . . , un−2 ). Unless otherwise stated a cyclic code is assumed to be linear. It is convenient to represent a codeword u = (u0 , u1 , . . . , un−1 ) by the polynomial f (x) = u0 + u1 x + · · · + un−1 xn−1 in the ring Fn [x] of polynomials modulo xn − 1 with coeﬃcients from F . Then xf (x) represents a cyclic shift of u, and so a linear cyclic code is represented by an ideal in Fn [x]. It is well-known from algebra that this ideal can be generated by a single polynomial g(x), a generator polynomial for the code. It is easily seen that g(x) divides xn − 1 over F and that the dimension of ℵ is k = n − deg g(x). To end this section, we introduce several important codes as examples. First of all, we observe a general principle for cyclic codes. Let n be relatively prime to q, and let ξ be a primitive nth root of unity, so that xn − 1 =

n−1 # i=0

x − ξi .

5.1. Codes

81

If l is the multiplicative order of q modulo n, then ξ ∈ F where F is a suitable ﬁnite ﬁeld of ql elements. Also, # x − ξj g(x) = j∈J

for some set J ⊆ {0, 1, . . . , n − 1}. Example 5.1 (Binary Hamming Codes). Let n = 2l − 1, k = n − l, d = 3, and J = 1, 2, 22, . . . , 2l−1 . Then the corresponding g(x) generates a binary Hamming code Hn over F = {0, 1}. Based on this code, the corresponding extended binary Hamming code Hn+1 is deﬁned by Hn+1

n+1 = (u1 , u2 , . . . , un+1 ) : (u1 , u2 , . . . , un ) ∈ Hn , ui = 0 . i=1

Example 5.2 (Quadratic Residue Codes). Let p and n be primes such that p is a square modulo n. For p = 2, the most important case, this means that n is a prime of the form 8l ± 1. Let F be the ﬁeld {0, 1, . . . , p − 1} and let J = {j : j = 0 is a square modulo n} . Then the corresponding g(x) generates a quadratic residue code over F . In particular, the binary Golay code G23 is the quadratic residue code over F = {0, 1} of length 23. Based on this code, the extended binary Golay code G24 is deﬁned by G24 =

24 (u1 , u2 , . . . , u24 ) : (u1 , u2 , . . . , u23 ) ∈ G23 , ui = 0 . i=1

Example 5.3 (Bose-Chaudhuri-Hocquenghem Codes). The BCH code Bn,d of length n and distance d over F is the cyclic code whose generator polynomial g(x) has roots exactly ξ, ξ2 , . . . , ξ d−1 , and their conjugates. Then, routine computation yields that d(Bn,d ) ≥ d. The special BCH codes over F of length q − 1, where q = card{F }, are usually called Reed-Solomon codes. In these cases, d(Bn,d ) = d.

82

5. Sphere Packings Constructed from Codes

Example 5.4 (Justesen Codes). Let R be a Reed-Solomon code over F , card{F } = 2l , with length n = 2l − 1, dimension k, and minimal Hamming distance d(R) = n − k + 1. Let ν be a primitive element of F . If u = (u0 , u1 , . . . , un−1 ) is a codeword of R, we deﬁne u∗ to be the vector u∗ = u0 , u0 , u1 , νu1 , . . . , un−1 , ν n−1 un−1 of length 2n over F . Since F is a vector space of dimension l over {0, 1}, we may set up a one-to-one correspondence between F and {0, 1}l . Let u be obtained from u∗ by replacing each component by the corresponding binary l-tuple. Then u is a binary vector of length 2ln. The linear code J = {u : u ∈ R} is called a Justesen code. It can be easily shown that its binary dimension is kl. Example 5.5 (Reed-Muller Codes). Let m(k) be the number of ones in the binary expansion of k. For 1 ≤ j ≤ l − 2, the jth-order binary punctured Reed-Muller code Mn of length n = 2l − 1 is the cyclic code whose generator polynomial has as roots those ξ k such that 1 ≤ k ≤ 2l − 2 and 1 ≤ m(k) ≤ l − j − 1, where ξ is a primitive element of F . Then, we deﬁne Mn+1 =

(u1 , u2 , . . . , un+1 ) : (u1 , u2 , . . . , un ) ∈ Mn ,

n+1

ui = 0

i=1

and call it the jth-order Reed-Muller code of length n + 1 = 2l .

5.2. Construction A Theorem 5.1 (Leech and Sloane [2]). Let ℵ be an (n, m, d) binary code, and let h(t) be the number of its codewords u such that u, oH = t. Deﬁne X = {x ∈ E n : x (mod 2) ∈ ℵ} ,

(5.1)

5.2. Construction A

83 √ r=

and

1

d/2 if d ≤ 4, if d ≥ 4,

 d if d < 4,  2 h(d) τ= 2n + 24 h(4) if d = 4,  2n if d > 4.

Then rSn + X is a periodic packing with density δ(rSn , X) =

mωn rn , 2n

in which the sphere rSn touches τ others. In particular, X is a lattice if and only if ℵ is a linear code. Suppose ℵ is a linear code of dimension k. Then the corresponding lattice sphere packing has density ωn r n δ(rSn , X) = n−k . 2 Proof. By its deﬁnition, X is a periodic discrete set and can be represented as X = ℵ + 2Zn , (5.2) where the codewords are regarded as points in E n . If d < 4, the points closest to the origin o are the 2d h(d) points of type [1d |0n−d ] obtained √ from the codewords of weight d. Such points are at Euclidean distance d from the origin. If d > 4, the points closest to the origin are the 2n points of type [21 |0n−1 ], at Euclidean distance 2. Finally, if d = 4, both sets of points are at the same distance 2 from the origin. Thus, by (5.2) and routine computation, rSn + X forms a periodic packing with density mωn rn δ(rSn , X) = , (5.3) 2n in which rSn touches τ others. Since ℵ is a code over F = {0, 1}, routine argument shows that X is a lattice if and only if ℵ is linear. When ℵ is a linear code of dimension k, m = 2k . Thus, the last assertion follows from (5.3). Theorem 5.1 is proved.

2

The construction deﬁned by (5.1) is usually called Construction A. Example 5.6. Let ℵ be a linear binary code with codewords u0 = (0, 0, 0), u1 = (0, 1, 1), u2 = (1, 0, 1), and u3 = (1, 1, 0). Then the lattice given by Construction A is the face-centered cubic lattice that attains both δ ∗ (S3 ) and k∗ (S3 ).

84

5. Sphere Packings Constructed from Codes

Example 5.7. Let ℵ be the extended Hamming code H8 deﬁned in Example 5.1. By Theorem 5.1 one can construct a lattice sphere packing S8 + Λ8 that attains both δ∗ (S8 ) and k∗ (S8 ). Example 5.8. In 1954, Golay [1] constructed a nonlinear (9, 20, 4) code. Applying Construction A to this code, Conway and Sloane [1] deduced that k(S9 ) ≥ 306. Thus, comparing with Theorem 2.6, we have k(S9 ) > k∗ (S9 ). Remark 5.1. With a (mod p) version of Construction A, Rush [1] was able to obtain δ ∗ (Sn ) ≥ 2−n(1+o(1)), which is asymptotically the Minkowski-Hlawka lower bound. As shown in Chapter 3, there are several proofs for the Minkowski-Hlawka theorem. However, none can be used to construct lattice sphere packings with such densities, since all the known proofs are based on mean value arguments. Rush’s method also shares this drawback, since the codes he has used cannot be clearly constructed.

5.3. Construction B Theorem 5.2 (Leech and Sloane [2]). Let ℵ be an (n, m, d) binary code with the property that the weight of each codeword is even, and let h(t) be the number of its codewords u such that u, oH = t. Deﬁne n n X = x ∈ E : x (mod 2) ∈ ℵ, xi ∈ 4Z , (5.4) i=1

√ √d/2 if d ≤ 8, r= 2 if d ≥ 8, and

 d−1 if d < 8,  2 h(d) τ= 2n(n − 1) + 27 h(8) if d = 8,  2n(n − 1) if d > 8.

Then rSn + X is a periodic packing with density δ(rSn , X) =

mωn rn , 2n+1

5.4. Construction C

85

and the sphere rSn touches τ others. In particular, X is a lattice if and only if ℵ is linear. Suppose ℵ is a linear code of dimension k. Then the corresponding lattice sphere packing has density ωn r n δ(rSn , X) = n+1−k . 2 This theorem can be proved in a way similar to Theorem 5.1, so its proof is omitted here. Usually, (5.4) is called Construction B. With this method one can construct some sphere packings with high densities. Here we give two examples. Example 5.9. Let ℵ be the ﬁrst-order Reed-Muller code M16 deﬁned in Example 5.5. By Theorem 5.2 one can construct a lattice Λ16 such that S16 + Λ16 is a packing with density ω16 δ(S16 , Λ16 ) = 4 2 in which each sphere touches 4320 others. This is the densest lattice sphere packing known in E 16 . Example 5.10. Let ℵ be the extended Golay code G24 deﬁned in Example 5.2 and let X be the set constructed by (5.4). Then the set Λ24 = √12 X u + √12 X , where u = √18 (1, 1, . . . , 1, −3), is a lattice known as the Leech lattice. This lattice has many important properties. For instance, up to isometry, S24 + Λ24 is the only packing in which every sphere touches k(S24 ) = k∗ (S24 ) = 196560 others (see Theorem 9.4). In addition, δ(S24 , Λ24 ) =

π 12 , 12!

which is the densest lattice sphere packing known in E24 .

5.4. Construction C Let x = (x1 , x2 , . . . , xn ) be a point in E n with integer coordinates. If the component xi of x can be expanded as a binary series xi =

∞ j=0

xji 2j ,

86

5. Sphere Packings Constructed from Codes

where complementary notation is used for negative integers, we call the matrix   x01 x02 · · · x0n  x11 x12 · · · x1n    A(x) =  x21 x22 · · · x2n    .. .. . . .. . . . . the coordinate array of x. For example, the coordinate array of (−2, −1, 1, 2) in E 4 is   0 1 1 0  1 1 0 1     1 1 0 0 .   .. .. .. .. . . . . For convenience, letting u and v be codewords of a binary code ℵ, we say that v is contained in u if ui = 1 whenever vi = 1. Theorem 5.3 (Leech and Sloane [2]). Suppose ℵ0 , ℵ1 , . . . , ℵk are binary codes of types (n, m0 , d0 ), (n, m1 , d1 ), . . . , (n, mk , dk ), respectively, where di = η4k−i and η = 1 or 2. Deﬁne X = {x ∈ Zn : (xj1 , xj2 , . . . , xjn ) ∈ ℵj , j = 0, 1, . . . , k} , r= and τ=

√

(5.5)

η 2k−1 ,

k

τj (u),

j=0 u∈ℵj

where τj (u) is the number of codewords in ℵj+1 ∩ ℵj+2 ∩ · · · ∩ ℵk that are contained in the codeword u of ℵj . Then rSn + X is a periodic packing with density k ωn η n/2 # δ(rSn , X) = mj 22n j=0 in which the sphere rSn touches τ others. Proof. Let x and y be distinct points in X, and assume that j is the ﬁrst index such that (xj1 , xj2 , . . . , xjn ) = (yj1 , yj2 , . . . , yjn ). We now consider two cases. Case 1. j > k. Then the Euclidean distance between x and y is at least 2k+1 .

5.4. Construction C

87

Case 2. 0 ≤ j ≤ k. Since ℵj is an (n, mj , dj ) code with dj = η4k−j , the two points x and y diﬀer in at least dj coordinates, and therefore the Euclidean distance between them is at least " √ dj 4j = η 2k . So in both cases we have min x, y ≥

x,y∈X x=y

√ k η2

and rSn + X is a periodic packing. As in the proof of Theorem 5.1, it is easy to see that there are m=

k #

mj

j=0

points of X in the cube I = x ∈ E n : 0 ≤ xi < 2k+1 . Thus, by the periodic property of X it can be deduced that the packing density of rSn + X is k mv(rSn ) ωn η n/2 # δ = (k+1)n = mj . 22n j=0 2

Let X be the set of points x ∈ X such that rSn + x touches rSn at its boundary, and let Xj be the set of points x ∈ X of [(2j )dj |0n−dj ] type. For a point x ∈ X, suppose that j is the ﬁrst index such that (xj1 , xj2 , . . . , xjn ) = (0, 0, . . . , 0). Then, by routine argument it follows that both (xj1 , xj2 , . . . , xjn ) and x are of [(2j )dj |0n−dj ] type if x ∈ X . Thus, we have card{X } =

k

card{Xj }.

j=0

By the deﬁnition of the coordinate array we have 0 if xj = 2j , xli = 1 if xj = −2j for all indices l ≥ j. Thus, if x ∈ X is a point such that (xj1 , xj2 , . . . , xjn ) ∈ ℵj ,

(5.6)

88

5. Sphere Packings Constructed from Codes

then the minus signs of its coordinates must be at the locations of the ones in some codeword v ∈ ℵj+1 ∩ ℵj+2 ∩ · · · ∩ ℵk . (5.7) On the other hand, for any codeword of (5.7) that is contained in a codeword u ∈ ℵj there is a point x ∈ X such that x, o = 2r. Thus, card{Xj } =

τj (u).

(5.8)

u∈ℵj

Then (5.6) and (5.8) together yield card{X } =

k

τj (u) = τ,

j=0 u∈ℵj

which implies the last assertion of Theorem 5.3.

2

Usually, (5.5) is called Construction C. Remark 5.2. For general codes, the set X constructed by (5.5) is not a lattice. However, if the codes ℵ0 , ℵ1 , . . . , ℵk are linear and nested, i.e., ℵj ⊆ ℵj+1 for all indices 0 ≤ j ≤ k − 1, then the set is a lattice. Example 5.11. Let ℵj be the 2jth Reed-Muller code of length n = 2m . In addition, take η = 1 and k = m/2 if m is even; and take η = 2 and k = (m − 1)/2 if m is odd. Then, by applying Construction C, Leech [2] was able to construct a lattice packing Sn + Λn in which every sphere touches m # 2 + 2i ∼ 4.768 . . . n(m+1)/2 τ= i=1

others. So far this is the best known lower bound for k∗ (Sn ) when n = 2m is large. Remark 5.3. Construction C can be used to obtain dense sphere packings in high dimensions. For example, by applying this construction to certain Justesen codes, Sloane [1] has shown that δ(Sn ) ≥ 2−6n(1+o(1)) for integers n of k2k type. This result is weaker than the MinkowskiHlawka theorem. Since the Justesen codes used cannot be clearly constructed, Sloane’s lower bound is not eﬃciently constructed.

5.5. Some General Remarks

89

5.5. Some General Remarks Besides the previous constructions, there are many other ingenious methods for constructing sphere packings. For example, applying ideas from algebraic curves, Litsyn and Tsfasman [1], [2] obtained a nonlattice packing Sn + X with δ(Sn , X) ≥ 2−1.31n(1+o(1)) and a lattice packing Sn + Λ with δ(Sn , Λ) ≥ 2−2.30n(1+o(1)). For more about sphere packing constructions, we refer to Conway and Sloane [1]. To end this chapter, based upon Example 5.11 we propose the following problems. Problem 5.1. Does there exist a positive number c such that k ∗ (Sn ) ≥ 3cn holds for every suﬃciently high dimension? If so, what is the largest c? Very probably the answer to this problem is “no.” If so, the following modiﬁcation would be interesting. Problem 5.2. Does there exist a positive number c such that k ∗ (Sn ) ≥ 3cn holds for inﬁnitely many dimensions? If so, what is the largest c?

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6. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres I

6.1. Blichfeldt’s Upper Bound for the Packing Densities of Spheres In 1929 H.F. Blichfeldt discovered a remarkable method for determining an upper bound for δ(Sn ). Roughly speaking, his idea runs as follows. Let Sn + X be a packing and let r be a number such that r > 1. Then, replace the spheres Sn + x, where x ∈ X, by rSn + x and ﬁll each of these new spheres with a certain amount of mass, of variable density, such that the total mass at any point of E n does not exceed 1. Hence, the total mass of the spheres rSn + x, where x ∈ X ∩ (l − r)In , does not exceed the volume of the large cube lIn . In this way, Blichfeldt obtained the ﬁrst signiﬁcant upper bound for δ(Sn ). Theorem 6.1 (Blichfeldt [1]). δ(Sn ) ≤

n + 1 −0.5n 2 . 2

In order to prove this theorem, we introduce two fundamental lemmas. Lemma 6.1. Let m be a positive integer and let x, x1 , . . . , xm be arbitrary points in E n . Then, one has m j, k=1

xj , xk 2 ≤ 2m

m j=1

x, xj 2 .

92

6. Upper Bounds for δ(Sn ) and k(Sn ), I

Proof. Let a1 , a2 , . . . , am be m real numbers. Routine analysis yields m

m 2 aj + a2k − 2aj ak

2

(aj − ak ) =

j, k=1

= ≤

j, k=1 m j, k=1 m

2 m 2 2 aj + ak − 2 aj j=1

2 aj + a2k

j, k=1 m

= 2m

a2j .

j=1

Thus, writing yj = x − xj = (yj1 , yj2 , . . . , yjn ), we have m

m

xj , xk 2 =

j, k=1

=

yj , yk 2

j, k=1 m n

2

(yji − yki )

i=1 j, k=1 m n

≤ 2m

(yji )2

i=1 j=1

= 2m

= 2m

n m

2

(yji )

j=1 i=1 m

yj 2

j=1

= 2m

m

x, xj 2 .

j=1

2

Lemma 6.1 is proved.

Lemma 6.2 (Blichfeldt [1]). Let r be a positive number, and let σ(x) be a function that is nonnegative and continuous for 0 ≤ x ≤ r and vanishes for x > r. Suppose that for each packing Sn + X, σ(x, xi ) ≤ 1 (6.1) xi ∈X

holds for every point x ∈ En . Then δ(Sn ) ≤ J −1 ,

6.1. Blichfeldt’s Upper Bound for δ(Sn ) %

where

93

r

xn−1 σ(x)dx.

J =n 0

Proof. Let l be a large number and suppose that there are exactly m spheres Sn + x1 , Sn + x2 , . . . , Sn + xm of the packing Sn + X that are entirely contained in the large cube lIn . It is obvious that rSn + xi ⊂ (l + 2r)In , and therefore σ(x, xi ) = 0

(6.2)

holds for every index i = 1, 2, . . . , m if x ∈ (l + 2r)In . Then, by (6.1) and (6.2) it follows that %

m

v((l + 2r)In ) ≥ =

σ(x, xi )dx

(l+2r)In i=1 m %

σ(x, xi )dx

i=1

%

En

=m

σ(x)dx n %Er

σ(x)d(ωn xn ) % r = mnωn xn−1 σ(x)dx =m

0

0

= mωn J. Hence, by the deﬁnition of δ(Sn ), we have mωn r→∞ v(lIn ) X v((l + 2r)In ) ≤ sup lim sup Jv(lIn ) r→∞ X

n 2r = sup lim sup 1 + J −1 l r→∞ X

δ(Sn ) = sup lim sup

= J −1 . Lemma 6.2 is proved.

2

With these preparations, we proceed to prove Theorem 6.1. Proof of Theorem 6.1. Let Sn + X be a packing and, for convenience, enumerate the points of X by x1 , x2 , . . . . Since Sn + X is a packing, we have xj , xk ≥ 2

94

6. Upper Bounds for δ(Sn ) and k(Sn ), I

whenever j = k and therefore, for each ﬁnite set of distinct indices i1 , i2 , . . . , im , m xij , xik 2 ≥ 4m(m − 1). j, k=1

Thus, for each point x ∈ En , by Lemma 6.1 we have m

x, xij ≥ 2(m − 1).

(6.3)

j=1

Deﬁning

one has r =

√

σ(x) = max 0, 1 − 12 x2 , 2. In addition, for every point x ∈ En , it follows that ∞

σ(x, xi ) =

i=1

=

m

σ(x, xij )

j=1 m j=1

1 1 − x, xij 2 2 1 x, xij 2 , 2 j=1 m

= m−

where ij are the indices i such that x, xi < ∞

√

2. Hence, by (6.3),

σ(x, xi ) ≤ m − (m − 1) = 1.

i=1

This means that the function σ(x) satisﬁes the condition of Lemma 6.2. Then, routine computation yields that √ 2

% J =n 0

1 2 xn−1 1 − x2 dx = 20.5n . 2 n+2

Thus, it follows from Lemma 6.2 that δ(Sn ) ≤ J −1 = Theorem 6.1 is proved.

n + 2 −0.5n 2 . 2 2

Remark 6.1. Blichfeldt’s method was improved by Rankin [1], [6], Sidelnikov [1], [2], Leven˘ s tein [1], [2], and others. In this way, Theorem 6.1 was improved to δ(Sn ) ≤ 2−0.5237n(1+o(1)).

6.2. Rankin’s Upper Bound for k(Sn )

95

By studying the number of spheres that can be packed in an n-dimensional torus, Yudin [1] gave a new proof of this result.

6.2. Rankin’s Upper Bound for the Kissing Numbers of Spheres Let m(n, α) be the maximal number of caps of geodesic radius α that can be packed in bd(Sn ). It is easy to see that k(Sn ) = m(n, π/6) and δ(Sn ) = lim

α→0

m(n + 1, α)s(α) , (n + 1)ωn+1

(6.4)

(6.5)

where s(α) indicates the area of a cap of geodesic radius α. Generalizing Blichfeldt’s method from En to bd(Sn ) and applying (6.4) and (6.5), R.A. Rankin obtained a new proof of Theorem 6.1. In addition, he obtained the following upper bound for k(Sn ). Theorem 6.2 (Rankin [6]). √ k(Sn )

πn3 0.5n 2 , 2

where f (x) g(x) means lim sup x→∞

f (x) ≤ 1. g(x)

First, let us introduce a set of fundamental lemmas. Lemma 6.3 (Rankin [6]). Let xj = (xj1 , xj2 , . . . , xjn ), j = 1, 2, . . . , m, be m distinct points in bd(Sn ), and deﬁne d=

min

1≤j
{xj , xk } .

Then, for any point x ∈ bd(Sn ), we have m j=1

where dj = x, xj .

2 d2j

− 4m

m j=1

d2j + 2m(m − 1)d2 ≤ 0,

96

6. Upper Bounds for δ(Sn ) and k(Sn ), I

Proof. We may suppose, without loss of generality, that x = (1, 0, . . . , 0). Then, by the deﬁnition of d we have n 1 2 m(m − 1)d ≤ 2 i=1

=

1≤j
n

2

(xji ) −

m

i=1

=m

(xji − xki )

m

j=1

m

2

2 xji

j=1

2 m 2 (1 − xj1 ) − m − xj1

j=1

j=1

2 n m m 2 + (xji ) − xji m =m

i=2 m

j=1

j=1 2

(1 − xj1 ) +

j=1

−

m

2 (1 − xj1 )

2

(xji )

i=2 n m

−

j=1

Since

n

i=2

2 xji

.

j=1

n 2 2 xji = 2 1 − xj1 , d2j = 1 − xj1 + i=2

it follows that m 2 m 1 1 2 m(m − 1)d2 ≤ m d2j − d , 2 4 j=1 j j=1 2 √ Lemma 6.4 (Rankin [6]). If 0 < β < π/2 and tanβ = o( n) for large n, then % β sec2 β (sin θ)n−2 (cos θ − cos β)dθ ∼ (sin β)n+1 . n2 0 which implies the lemma.

Proof. Substituting t = (n − 1) log

sin β , sin θ

we have % β % − 12 −2t (sin β)n−1 ∞ n−2 cos β (sin θ) dθ = 1 + 1 − e n−1 tan2 β e−t dt. n−1 0 0 (6.6) By Taylor’s theorem it follows that (1 + x)−1/2 = 1 − 12 x + 38 η(x)x2 ,

(6.7)

6.2. Rankin’s Upper Bound for k(Sn )

97

where 0 ≤ η(x) ≤ 1 for all x ≥ 0. Thus, taking −2t x = 1 − e n−1 tan2 β and substituting (6.7) in (6.6) we obtain, for a corresponding η(t) satisfying 0 ≤ η(t) ≤ 1, % β % −2t (sin β)n−1 ∞ −t 1 n−2 cos β (sin θ) dθ = e 1− 1 − e n−1 tan2 β n−1 2 0 0 2 −2t 3 4 n−1 + η(t) 1 − e tan β dt 8 (sin β)n−1 tan2 β 3η tan4 β = 1− + , n−1 n+1 (n + 1)(n + 3) √ where 0 ≤ η ≤ 1. Consequently, since tanβ = o( n), we have

% β 3η tan2 β sec2 β (sin β)n+1 1 − (sin θ)n−2 (cos θ − cos β)dθ = 2 n −1 n+3 0 2 sec β ∼ (sin β)n+1 . n2 Lemma 6.4 is proved. 2 √ Lemma 6.5 (Rankin [6]). If 0 < α < π/4 and β = arcsin( 2 sin α), then √ π Γ((n − 1)/2) sin β tanβ m(n, α) ≤ . +β 2 Γ(n/2) 0 (sin θ)n−2 (cos θ − cos β)dθ Proof. Let α be a positive number with α < π/4, and suppose that there are m(n, α) nonoverlapping caps Ωi (α) of geodesic radius α on bd(Sn ). We replace each cap Ωi (α) by a concentric cap Ωi (β) of geodesic radius β, where √ β = arcsin( 2 sin α). Also, letting γ = 2 sin β2 be the Euclidean radius of Ωi (β), we attach a mass density cos β 2 2 cos β 2 (cos θ − cos β) σi (r) = 2 (γ − r ) = sin β sin2 β with respect to Ωi (β) to the points x ∈ bd(Sn ), where θ r = x, xi = 2 sin . 2 Then, the total mass of Ωi (β) is % β θ (n − 1)ωn−1 (sin θ)n−2 σ(2 sin )dθ J = 2 0 % β cos β = 2(n − 1)ωn−1 2 (sin θ)n−2 (cos θ − cos β)dθ. sin β 0

(6.8)

98

6. Upper Bounds for δ(Sn ) and k(Sn ), I

Let x be a point of bd(Sn ). Then either x belongs to no cap Ωi (β), in which case the total mass density at x is zero, or else x belongs to m such caps centered at x1 , x2 , . . . , xm , say. In the latter case, the total mass density at x is, in the notation of Lemma 6.3, m

σ(x) =

σi (di ) =

i=1

m cos β 2 2 mγ − di . sin2 β i=1

Thus m

β − σ(x) sin β tanβ 2 β 1 + sec β = 4 m− σ(x) sin2 , 2 2

d2i = 4m sin2

i=1

and so, by Lemma 6.3, 2 1 + sec β 1 + sec β β β 0 ≥ 16 m − σ(x) sin4 − 16m m − σ(x) sin2 2 2 2 2 + 4m(m − 1) sin2 β. This reduces to 4m(1 − σ(x)) ≥ σ(x)2 tan2 β, which implies σ(x) ≤ 1. Therefore, we deduce that m(n, α)J is less than the total surface area of Sn and so, by (6.8), √ nωn π Γ((n − 1)/2) sin β tanβ m(n, α) ≤ = . +β J 2 Γ(n/2) 0 (sin θ)n−2 (cos θ − cos β)dθ Lemma 6.5 is proved.

2

With these preparations, Theorem 6.2 can be proved as follows. Proof of Theorem 6.2. It follows from Lemma 6.4 and Lemma 6.5 that √ πn3 sin 2β m(n, α) √ . 2( 2 sin α)n Therefore, by (6.4), √ πn3 0.5n k(Sn ) = m(n, π/6) 2 . 2 Theorem 6.2 is proved.

2

6.3. An Upper Bound for δ(Sn (α))

99

√ Since β = arcsin( 2 sin α), by (6.5), Lemma 6.4, and Lemma 6.5 it follows that +α m(n + 1, α) 0 nωn (sin θ)n−1 dθ δ(Sn ) = lim α→0 (n + 1)ωn+1 +α 2 sin β 0 (sin θ)n−1 dθ ≤ lim + β α→0 2 (sin θ)n−1 (cos θ − cos β)dθ 0 +α (n + 1)2 0 (sin θ)n−1 dθ = lim α→0 2(sin β)n 2 (n + 1) (sin α)n = lim α→0 2n(sin β)n n + 2 −0.5n ∼ 2 . 2 Thus, in asymptotic form, we obtain a new proof for Theorem 6.1. Remark 6.2. To determine the exact value of m(n, α) for some special n and α is a well-known and very challenging problem. It was proved by Davenport and Haj˘ os [1] that n + 1 if π4 < α ≤ π4 + 12 arcsin n1 , m(n, α) = 2n if α = π4 . This result implies a remarkable fact: If n + 2 congruent caps can be packed into bd(Sn ), then so can 2n.

6.3. An Upper Bound for the Packing Densities of Superspheres Applying Blichfeldt’s method to superspheres Sn (α), we obtain the following theorem. Theorem 6.3 (van der Corput and Schaake [1], Hlawka [2], Hua [1], and Rankin [2] and [3]). Writing β = 1/α, ) 1+βn if α ≥ 2, 2βn δ(Sn (α)) ≤ 1+(1−β)n if 1 ≤ α ≤ 2. 2βn It is easy to see that Theorem 6.3 implies Theorem 6.1, since Sn = Sn (2). In order to prove this theorem, we introduce a generalization of Lemma 6.1.

100

6. Upper Bounds for δ(Sn ) and k(Sn ), I

Lemma 6.6 (Hua [1]). Let a1 , a2 , . . . , am be m real numbers. Then  m

  m |ai |α if α ≥ 2,  2α−1m α i=1 |ai − aj | ≤ m

  |ai |α if 1 ≤ α ≤ 2. i, j=1  2m i=1

Proof. Deﬁning f (γ) = max

m

Σ|ai |=0

|ai − aj |1/γ

γ

−γ m m |ai |1/γ ,

i, j=1

(6.9)

i=1

it follows from a well-known analytical theorem of Riesz [1] (see also Theorem 296 of Hardy, Littlewood, and P´ olya [1]) that log2 f (γ) is a convex function of γ for 0 ≤ γ ≤ 1. Routine computation yields that f (0) = max max{|ai − aj |}/ max{|ai |} = 2, Σ|ai |=0

i, j

i

,  &

2 '

−1  m m m  . 2m a2i − 2 ai a2i f ( 12 ) = max m  Σ|ai |=0  i=1 i=1 i=1 √ = 2, and

)

−1 2 m m 2(m − 1) |ai | m |ai | ≤ 2. f (1) ≤ max Σ|ai |=0

i=1

i=1

Therefore, we have ) log2 f (γ) ≤

1−γ

if 0 ≤ γ ≤ 12 ,

γ

if

1 2

≤ γ ≤ 1.

(6.10)

Substituting α = 1/γ, it follows from (6.9) and (6.10) that  m

  m |ai |α if α ≥ 2,  2α−1 m α i=1 |ai − aj | ≤ m

  |ai |α if 1 ≤ α ≤ 2. i, j=1  2m i=1

2

Lemma 6.6 is proved. n

For an arbitrary point x = (x1 , x2 . . . , xn ) of E , we deﬁne xα =

n i=1

1/2 |xi |α

.

6.4. Hans Frederik Blichfeldt

101

Meanwhile, we deﬁne a density function √ 1 − 12 x2 if 0 ≤ x ≤ 2, σ(x) = 0 otherwise. Suppose that Sn (α) + X is a packing, and suppose that there are m points of X, say x1 , x2 , . . . , xm , such that √ x − xj α < 2, j = 1, 2, . . . , m. Since

n

|xji − xki |α ≥ 2α ,

j = k,

i=1

by Lemma 6.6 and routine calculation it follows that the mass density at x satisﬁes σ(x) =

m

σ(x − xj α )

j=1

1 |xi − xji |α ≤ 1. 2 j=1 i=1 m

= m−

n

Then, Theorem 6.3 can be proved by an argument that is similar to the proof of Lemma 6.2. We omit the details. Remark 6.3. By applying more eﬃcient density functions, the results of Hlawka [2], [3] and Rankin [2], [3] are slightly better than Theorem 6.3.

6.4. Hans Frederik Blichfeldt Hans Frederik Blichfeldt was born in Denmark in 1873. In 1888 he took, and passed with high honors, the general preliminary examination conducted by the University of Copenhagen. In the same year, his family emigrated to the United States. During his early American years the young Blichfeldt found employment in Nebraska, Wyoming, Oregon, and Washington, where he worked in the lumber industry. Then for the two years 1892–1894 he worked as a draftsman for the engineering department of the City and County of Whatcom, Washington. In 1894, with an enthusiastic recommendation from the county superintendent of schools, Blichfeldt was admitted as a special student to Stanford University. There he received a bachelor’s degree in mathematics in 1896, followed by a master’s degree in 1897. At that time, German universities

102

6. Upper Bounds for δ(Sn ) and k(Sn ), I

were important centers of mathematical activity. So, with the help of Rufus Green, Blichfeldt went to Leipzig to work with Sophus Lie. In 1898 he was awarded a doctorate with a dissertation On a Certain Class of Groups of Transformations in Space of Three Dimensions. Blichfeldt returned to Stanford as an instructor in mathematics and remained a member of Stanford’s faculty until his retirement in 1938. He was assistant professor of mathematics, 1901–1906; associate professor, 1906– 1913; professor, 1913–1938; and professor emeritus from 1938 until his death in 1945. He was also associate professor of mathematics at the University of Chicago, summer quarter 1911, and professor of mathematics at Columbia University for the summer sessions of 1924 and 1925. As the head of the department from 1927 until 1938, he played a very important role in the development of Stanford into a leading mathematical center. Besides Theorem 2.4 and Theorem 6.1 of this book, Blichfeldt made important contributions to the geometry of numbers and group theory. For example, he solved the problem of ﬁnding all ﬁnite collineation groups in four variables, a problem whose solution had eluded Klein and Jordan. He published some two dozen research papers of importance. In addition, he was the author or coauthor of three books in group theory. In 1920, Blichfeldt was elected to the National Academy of Sciences of the United States and was once the vice-president of the American Mathematical Society. For his distinguished scientiﬁc contributions, he was awarded a Dannebrogorden in 1938 by the king of Denmark.

7. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres II

7.1. Rogers’ Upper Bound for the Packing Densities of Spheres Let T = conv{v0 , v1 , . . . , vn } be a regular simplex in E n of side length 2. We deﬁne

n i=0 v(T ∩ (Sn + vi )) σn = v(T ) (n + 1)v(T ∩ (Sn + v0 )) = . v(T ) In other words, σn is the ratio of the volume of the part of the simplex covered by the unit spheres centered at its vertices to the volume of the whole simplex. Using geometric methods, C.A. Rogers improved Blichfeldt’s upper bound in 1958 with the following result. Theorem 7.1 (Rogers [10]). δ(Sn ) ≤ σn ∼

n −0.5n 2 . e

This theorem is comparatively easy to imagine, but very diﬃcult to prove, requiring a detailed study of the Dirichlet-Voronoi cells. First, let us recall some basic facts about the Dirichlet-Voronoi cells. Suppose that

104

7. Upper Bounds for δ(Sn ) and k(Sn ), II

X = {x1 , x2 , . . . } is a discrete set of points in E n such that Sn + X is a packing and (Sn + X) ∩ int(Sn + x) = ∅ (7.1) for every point x ∈ E n . As usual, we denote the Dirichlet-Voronoi cell associated to xj by D(xj ). It is well-known (see Section 1.1) that D(xj ) is a convex polytope, Sn + xj ⊂ D(xj ), and the family of Dirichlet-Voronoi cells D(xj ) together form a tiling of E n . Also, it follows from (7.1) that d(D(xj )) < 4,

xj ∈ X.

Without loss of generality, we may assume that xj = o and consider the Dirichlet-Voronoi cell D(o). For any sequence F0 ⊂ F1 ⊂ · · · ⊂ Fn−1 of its faces such that dim{Fi } = i there is a simplex T with vertices o, v1 , . . . , vn such that vi ∈ Fn−i and vi , o = min {x, o} . x∈Fn−i

(7.2)

1 2 k If Fn−i−1 , Fn−i−1 , . . . , Fn−i−1 are the (n − i − 1)-dimensional faces of 1 2 }, conv{vi , Fn−i−1 }, . . . , Fn−i , then Fn−i is dissected into conv{vi , Fn−i−1 k conv{vi , Fn−i−1 }. Thus, by considering all such sequences of faces, the cell D(o) is dissected into a family of simplices T1 , T2 , . . . , Tl .

Now we introduce a fundamental lemma about the structure of these simplices. Lemma 7.1. Let T = conv{o, v1 , . . . , vn } be a simplex determined by (7.2). Then 2i vi , vj ≥ i+1 for 1 ≤ i ≤ j ≤ n. Proof. Since vi belongs to an (n − i)-dimensional face Fn−i , which is a subset of at least i facets of D(o), by the deﬁnition of the Dirichlet-Voronoi cell there are i points of X, say x1 , x2 , . . . , xi , such that xk − vi , xk − vi = vi , vi ,

k = 1, 2, . . . , i.

(7.3)

For convenience, we write x0 = o. Since Sn + X is a packing and therefore xk , xl ≥ 2 for k = l, by (7.3) it follows that 2i(i + 1) = 4≤ xk − xl , xk − xl 0≤k≤l≤i

0≤k≤l≤i

7.1. Rogers’ Upper Bound for δ(Sn )

= (i + 1)

105

i xk − vi , xk − vi k=0

−

i

(xk − vi ),

k=0

≤ (i + 1) = (i +

i

(xk − vi )

k=0

i

xk − vi , xk − vi

k=0 1)2 vi , vi .

Thus, we have vi , vi ≥

2i , i+1

i = 1, 2, . . . , n.

(7.4)

When 1 ≤ i < j ≤ n, since vi + θ(vj − vi ) ∈ Fn−i for 0 ≤ θ ≤ 1, it follows from (7.2) that vi + θ(vj − vi ), vi + θ(vj − vi ) ≥ vi , vi . Thus, we have 2θvi , vj − vi ≥ −θ2 vj − vi , vj − vi

(7.5)

for 0 ≤ θ ≤ 1. Since (7.5) holds for any arbitrarily small positive number θ, we deduce that vi , vj − vi ≥ 0. (7.6) Then, it follows from (7.4) and (7.6) that vj , vi = vi , vi + vi , vj − vi ≥ vi , vi ≥

2i . i+1 2

Lemma 7.1 is proved. Proof of Theorem 7.1. We suppose that δ(Sn ) > σn

and proceed to deduce a contradiction. By the deﬁnition of δ(Sn ) it follows that there is a periodic packing Sn + X such that δ(Sn , X) > σn . Then, there is a point xj ∈ X that v(Sn + xj ) > σn . v(D(xj ))

(7.7)

106

7. Upper Bounds for δ(Sn ) and k(Sn ), II

For convenience, we assume that xj = o. Suppose that D(o) is dissected into a family of simplices T1 , T2 , . . . , Tl as described in the paragraph preceding Lemma 7.1. Then (7.7) is equivalent to

l

k=1 v(Tk ∩ Sn )

l k=1 v(Tk )

> σn ,

which implies v(T ∩ Sn ) > σn , v(T )

(7.8)

where T = Tk for a certain index k. For convenience, we assume that T = conv{o, v1 , . . . , vn }, which satisﬁes (7.2). √ with vertices ( 2, 0, . . . , 0), (0, √ Let T be the n-dimensional simplex √ 2, 0, . . . , 0), . . . , (0, 0, . . . , 0, 2) in E n+1 , and let H be the hyperplane determined by these points. In a similar way to the process of dissecting D(o) into simplices T1 , T2 , . . . , Tl , one can dissect T into (n + 1)! simplices T1 , T2 , . . . , T(n+1)! , each of which is congruent with ∗ ∗ ∗ ∗ T = conv{v0 , v1 , . . . , vn }, where vi∗ =

√

√

2/(i + 1), 0, . . . , 0 .

2/(i + 1), . . . ,

It is easy to verify that if 1 ≤ i ≤ j ≤ n, then vi∗ − v0∗ , vj∗ − v0∗ =

2ij + 2i 2i = . (i + 1)(j + 1) i+1

Hence, by Lemma 7.1, vi , vj ≥ vi∗ − v0∗ , vj∗ − v0∗

(7.9)

for 1 ≤ i ≤ j ≤ n. Let L be the linear transformation from E n to H determined by n

λi vi → v0∗ +

i=1

n

λi (vi∗ − v0∗ ).

i=1

Clearly, L(T ) = T ∗ . In addition, L(Sn ) = E is an ellipsoid in H centered at v0∗ . Now, if x is a point of T ∩ Sn , we have x=

n

λi vi ,

i=1

where λi ≥ 0, and x, x ≤ 1.

(7.10)

7.2. Schl¨ aﬂi’s Function

107

Hence, by (7.9) and (7.10), the corresponding point y = L(x) ∈ T ∗ ∩ E satisﬁes y − v0∗ , y − v0∗ =

n n

λi λj vi∗ − v0∗ , vj∗ − v0∗

i=1 j=1

≤

n n

λi λj vi , vj

i=1 j=1

= x, x ≤ 1, which implies

T ∗ ∩ E ⊆ T ∗ ∩ (Sn + v0∗ ).

(7.11)

Since L preserves the ratio of two volumes, by (7.8) and (7.11) it follows that σn <

v(T ∩ Sn ) v(T ∗ ∩ E) v(T ∗ ∩ (Sn + v0∗ )) = ≤ . v(T ) v(T ∗ ) v(T ∗ )

(7.12)

On the other hand, it follows from the deﬁnition of σn and the fact that all simplices Ti are congruent with T ∗ that v(T ∗ ∩ (Sn + v0∗ )) = σn , v(T ∗ ) which contradicts (7.12). This contradiction shows that δ(Sn ) ≤ σn . Then, Theorem 7.1 follows from Daniel’s asymptotic formula (see Corollary 7.1). 2 Remark 7.1. Rogers’ upper bound is better than Blichfeldt’s only up to a certain constant. However, Rogers’ method reveals the geometric nature of the problem.

7.2. Schl¨ aﬂi’s Function To study the spherical geometry, Schl¨ aﬂi [1] introduced a remarkable function 2n s(n, α) Fn (α) = , n!nωn where s(n, α) is the area of a regular spherical simplex in bd(Sn ) of dihedral angle 2α. This function has been studied by many authors. To give a

108

7. Upper Bounds for δ(Sn ) and k(Sn ), II

general impression of this geometric function, we list a few of its well-known properties without proof. Fn ( 12 arcsec(n − 1)) = 0, Fn (α) + Fn (π − α) = and

2 Fn (α) = π

%

2n , n!

α 1 2 arcsec(n−1)

Fn−2 (β)dθ,

where sec 2β = sec 2θ − 2 and F0 (α) = F1 (α) = 1. In this section we prove the following asymptotic formula. Lemma 7.2 (Rogers [12]). Let α be a positive number such that α < π/4 −1 and sec 2α − n + 1 is bounded. Then, writing c = sec 2α − (n − 1) , √

n/2 1 + cn 2e Fn (α) ∼ √ . 2 e1/c n! πcn Proof. In E n , take q = (q, q, . . . , q), p1 = (p, 0, . . . , 0), p2 = (0, p, . . . , 0), . . . , pn = (0, 0, . . . , 0, p) such that p > 0, q < 0, pi ∈ bd(Sn ) + q,

i = 1, 2, . . . , n,

and the dihedral angle of the regular spherical simplex T = p1 p2 · · · pn in bd(Sn ) + q is 2α. For convenience, we write n V = q+ λi (pi − q) : λi ≥ 0 . i=1

Since

and

+ − x,q 2 e dx s(n, α) +V = − x,q 2 dx nωn e En % En

e− x,q dx = 2

nωn Γ(n/2) , 2

it follows that 2n+1 Fn (α) = n!nωn Γ(n/2)

%

e− x,q dx. 2

(7.13)

V

Then, applying a linear transformation L determined by L(q) = o and L(pi ) = pi /p for i = 1, 2, . . . , n, it follows from (7.13) that √ % % ∞ 2n 1 + cn ∞ Fn (α) = · · · e−Q(y) dy1 · · · dyn , (7.14) π n/2 n! 0 0

7.2. Schl¨ aﬂi’s Function

109

where Q(y) =

n

yi2

2 n +c yi .

i=1

i=1

When 2α is acute and c is positive, we can apply the well-known result % ∞ √ 2 −cy 2 πc e = e−w /c+2wyidw −∞

n

with y = i=1 yi to express the integrand of (7.14) in a factorizable form. Then, substituting w = u + vi with v = cn/2, we obtain √

% ∞ % ∞ % 2n 1 + cn ∞ −Q(y,w) ··· e dw dy1 · · · dyn Fn (α) = n/2 √ π n! cn 0 0 −∞ √ % 2n 1 + cn en/2−1/c ∞ (1−u2 )/c−nui/v = √ e Ψ(u, v)n du, (7.15) πc (2πcn)n/2 n! −∞ where Q(y, w) =

n

yi2 + w2 /c − 2wi

i=1

yi

i=1

%

and

n

∞

Ψ(u, v) = 2v

e−y

2

−2(v−ui)y

dy.

0

Now we can get an asymptotic expansion for Ψ(u, v) in inverse powers of v using integration by parts. Since the ﬁrst term of the expansion is 1, we can use the logarithmic series to convert this into an asymptotic expansion for log Ψ(u, v). The ﬁrst terms of this expansion take the form ui u2 + 1 − . v cn When we substitute the asymptotic expansion for log Ψ(u, v) in the formula

2 1 − u2 nui e(1−u )/c−nui/v Ψn (u, v) = exp − + n log Ψ(u, v) c v and apply the exponential expansion, we obtain an expansion for the in2 tegrand of (7.15) as a product of e−2u /c with an asymptotic expansion in inverse powers of v. Integrating this expansion term by term, we obtain the following formula: √

n/2 2e 1 + cn √ Fn (α) ∼ Φn (α), (7.16) 2 e1/c n! πcn where Φn (α) =

∞ 1 l [nΠ(, c)] , l! l=0

110

7. Upper Bounds for δ(Sn ) and k(Sn ), II

2 + 4 1 i + Π(, c) = − √ 4cn 2cn

r/2 ∞ ∞ t−1 (−1) & 1 + ir t 2cn t=1 r=1

and m =

0≤s≤r/2

0 cm/2 (m − 1)(m − 3) · · · 3

't r! r−2s , s!(r − 2s)!

if m is odd, if m is even.

As n tends to inﬁnity, since 1/c is bounded, it is easy to see that the asymptotic expansion is in powers of 1/n, and that it is valid, in the sense that the error made in omitting the terms after that in 1/nh is of order 1/nh+1 . Thus,

1 12 18 1 1 Φn (α) = 1 + (7.17) 1+ + 2 +O 2 . 12 c c n n Then, (7.16) and (7.17) together yield Lemma 7.2.

2

Remark 7.2. For an alternative proof of this formula we refer to B¨ or¨ oczky Jr. and Henk [2]. Corollary 7.1 (Daniel’s Asymptotic Formula). σn ∼

n −0.5n 2 . e

Proof. Let T = conv{o, v1 , v2 , . . . , vn } be a regular simplex in E n with side length 2, and denote its dihedral angle by 2α. Then, σn =

(n + 1)s(n, α) (n + 1)!nωn Fn (α) = . nv(T ) 2n nv(T ) √ 2n/2 n + 1 v(T ) = n!

It is well-known that

and

−1 c = sec 2α − (n − 1) = 1.

Thus, by Lemma 7.2 and Stirling’s formula, σn ∼ √

(n + 1)!en/2−1 n ∼ 2−0.5n . e 2 Γ(n/2 + 1)(4n)n/2 2

Corollary 7.1 is proved. Corollary 7.2. Let τn be the number deﬁned in Section 3.3. Then n τn ∼ √ . e e

7.3. The Coxeter-B¨ or¨ oczky Upper Bound for k(Sn )

111

Proof. It is easy to see that τn =

2n n+1

n/2 σn .

Therefore, Corollary 7.2 follows immediately from Corollary 7.1.

2

7.3. The Coxeter-B¨ or¨ oczky Upper Bound for the Kissing Numbers of Spheres 3 be the Let x, yg be the geodesic metric deﬁned in bd(Sn ), and let xy spherical segment from x to y. For a positive number φ < π/4 and a point x ∈ bd(Sn ), let Ω(x, φ) be the cap of geodesic radius φ and center x, and write µ(n, φ) = s(Ω(x, φ)). Let T = v1 v2 . . . vn be a regular spherical simplex in bd(Sn ) of geodesic side length 2φ. In a similar way to the number σn , we deﬁne

n s(Ω(vi , φ) ∩ T ) σn (φ) = i=1 s(T ) n s(Ω(v1 , φ) ∩ T ) = . s(T ) Also, let m(n, φ) be the maximal number of nonoverlapping caps of geodesic radius φ that can be packed into bd(Sn ). As an analogue of Theorem 7.1, H.S.M. Coxeter in 1961 made the following conjecture. Conjecture 7.1 (Coxeter [4]). m(n, φ) ≤

nωn σn (φ) 2Fn−1 (α) = , µ(n, φ) Fn (α)

(7.18)

aﬂi’s functions and where Fn−1 (α) and Fn (α) are Schl¨ sec 2α = sec 2φ + n − 2. This conjecture was proved by B¨ or¨ oczky [1]. As a consequence of (7.18) and Lemma 7.2, since k(Sn ) = m(n, π/6), (7.19) one can deduce the following upper bound for the kissing numbers of spheres.

112

7. Upper Bounds for δ(Sn ) and k(Sn ), II

Theorem 7.2 (B¨ or¨ oczky [1] and Coxeter [4]). √ πn3 k(Sn ) √ 20.5n . e 2 B¨ or¨ oczky’s proof of (7.18) is elementary but rather technical. Let X = {x1 , x2 , . . . , xm } be a discrete subset of bd(Sn ) such that the caps Ω(xi , φ), i = 1, 2, . . . , m, together form a packing in bd(Sn ). Analogously to the Euclidean case, the spherical Dirichlet-Voronoi cell is deﬁned by D (xi ) = {x ∈ bd(Sn ) : x, xi g ≤ x, xj g , j = 1, 2, . . . , m} . In other words, D (xi ) =

m 4

Hij ,

j=1

where Hij = {x ∈ bd(Sn ) : x, xi − xj ≥ 0} . From the deﬁnition of D (xi ), it is easy to see that Ω(xi , φ) ⊂ D (xi ) for i = 1, 2, . . . , m. Thus, if we can prove µ(n, φ) ≤ σn (φ) s(D (xi )) for all indices i = 1, 2, . . . , m, we can deduce (7.18) immediately. In bd(Sn ), n ≥ 3, let gi (φ) be the geodesic distance from the center of an i-dimensional regular spherical simplex of side length 2φ to its vertices, and let g(φ) be the geodesic distance from the center of a regular spherical cross polytope of side length 2φ to its vertices. It is easy to see that φ = g1 (φ) < g2 (φ) < · · · < gn−1 (φ) < g(φ). Let X = {x1 , x2 , . . . , xm(n,φ) } be a discrete set of points in bd(Sn ) such that the caps Ω(xi , φ), i = 1, 2, . . . , m(n, φ), form a cap packing in bd(Sn ). We introduce two simple results concerning the geometry of the spherical Dirichlet-Voronoi cells D (xi ). Lemma 7.3. Let Fn−i−1 be an (n − i − 1)-dimensional common face of the i + 1 spherical Dirichlet-Voronoi cells D (x1 ), D (x2 ), . . . , D (xi+1 ). Let Sn−i−1 be the spherical subspace of bd(Sn ) determined by Fn−i−1 . Then, for j = 1, 2, . . . , i + 1, min

x∈Sn−i−1

x, xj g ≥ gi (φ),

7.3. The Coxeter-B¨ or¨ oczky Upper Bound for k(Sn )

113

where equality holds if and only if the i + 1 caps Ω(xj , φ) mutually touch one another. Proof. Let x be an arbitrary point of Sn−i−1 . Then the relations

x, xj g = ϕ,

j = 1, 2, . . . , i + 1,

∗ hold simultaneously for a certain positive number ϕ. Let Si+1 be the spherical subspace determined by the i + 1 points x1 , x2 , . . . , xi+1 . Observing ∗ the distribution of the i + 1 points in the spherical cap Si+1 ∩ Ω(x, ϕ), it follows from routine argument that

ϕ ≥ gi (φ), where equality holds if and only if the i + 1 caps Ω(xj , φ) mutually touch one another. Thus, Lemma 7.3 is proved. 2 Lemma 7.4. Let Fn−i−1 be an (n − i − 1)-dimensional common face of i + 1 spherical Dirichlet-Voronoi cells D (x1 ), D (x2 ), . . . , D (xi+1 ); let Sn−i−1 be the spherical subspace determined by Fn−i−1 ; and let S be the spherical subspace {x ∈ bd(Sn ) : x, x1 g = x, xi+2 g } . Suppose that y is a point of Sn−i−1 such that

x1 , yg =

min

x∈Sn−i−1

x1 , xg

and suppose that x5 1 y ∩ S = ∅. Then

x1 , yg ≥ g(φ). Proof. Let Sj∗ be the spherical subspace determined by x1 , x2 , . . . , xj . ∗ Obviously, y ∈ Si+1 . For convenience, we write ∗ Ω = Ω(y, x1 , yg ) ∩ Si+2 .

It is easy to see that the i + 1 points x1 , x2 , . . . , xi+1 lie on the relative boundary of the cap Ω. Since x5 1 y ∩ S = ∅, it follows from Figure 7.1 that xi+2 ∈ Ω. Let xi+2 be a point on the relative boundary of Ω such that ∗ 5 xi+2 xi+2 is orthogonal to Si+1 . Then we have xj , xi+2 g ≥ xj , xi+2 ≥ 2φ for j = 1, 2, . . . , i + 1.

114

7. Upper Bounds for δ(Sn ) and k(Sn ), II

x1 S

y xi+2 Ω

xi+2

Figure 7.1 Now, the i + 2 points x1 , x2 , . . . , xi+1 , and xi+2 lie on the boundary ∗ of a hemicap of Ω bounded by Si+1 . From elementary geometry we know that in this situation there are two points, x1 and x2 say, such that the 52 is less than or equal to π/2. Thus, spherical angle between x5 1 y and yx we have x1 , yg ≥ g(φ). 2

Lemma 7.4 is proved. vi+1

vi−1 u vi−2

vi u∗ Figure 7.2

3 and y 3z are In bd(Sn ), as usual, we say that the spherical segments xy orthogonal at y if their projections onto the tangent hyperplane of bd(Sn ) at y are orthogonal. Then, a spherical simplex T = v1 v2 . . . vn , where v1 , vi g < π/2 for i = 1, 2, . . . , n, is called an orthosimplex if v5 i vj and v5 j vk are pairwise orthogonal whenever 1 ≤ i < j < k ≤ n. More generally, 5vi , we call the spherical simplex if T is an orthosimplex and u ∈ vi−1 Tn,i,u = v1 . . . vi−1 uvi+1 . . . vn

7.3. The Coxeter-B¨ or¨ oczky Upper Bound for k(Sn )

115

a coorthosimplex. It is easy to see that the coorthosimplex can be obtained from another orthosimplex T ∗ = vn . . . vi+1 u∗ vi−1 . . . v1 such that u ∈ u∗5 vi+1 (see Figure 7.2). For convenience, we call u∗ and u∗∗ = vi the lower orthopoint and upper orthopoint of the vertex u of Tn,i,u , respectively. Clearly, vi∗ = vi∗∗ = vi if T is an orthosimplex. In order to prove a basic result about orthosimplices, we introduce a couple of basic properties of coorthosimplices. Assertion 7.1. Let p be a point of Tn,i,u such that p ∈ v5 i−1 u, and let p and p be the lower orthopoint and upper orthopoint of the vertex p of the subcoorthosimplex v1 . . . vi−2 pvi+1 . . . vn , respectively. When we move p along v5 i−1 u from vi−1 to u, both v1 , p g and v1 , p g increase. Proof. We examine only the v1 , p g case. u∗∗

p2 p1

p2

u vi+1

p1

vi−1

vn v1 Figure 7.3 When i ≤ 2 or i = n, the assertion is trivial. Otherwise, let p1 and p2 be two distinct points of the spherical segment v5 i−1 u such that vi−1 , p1 g < vi−1 , p2 g .

(7.20)

By elementary arguments, it can be shown that p∗j = pj ,

j = 1, 2.

(7.21)

Then, it follows from the deﬁnition of the lower orthopoint that vi−1 p1 vi+1 and vi−1 p2 vi+1 are right spherical triangles lying in the spherical subspace 5 S determined by vi−1 , u∗∗ , and vi+1 , with common hypotenuse vi−1 vi+1 (see Figure 7.3). Therefore, it follows from (7.20) and (7.21) that vi−1 , p1 g < vi−1 , p2 g . Observe that v15 vi−1 is orthogonal to S. Then (7.22) implies v1 , p1 g < v1 , p2 g .

(7.22)

116

7. Upper Bounds for δ(Sn ) and k(Sn ), II 2

Thus, the considered case is proved.

Assertion 7.2. Let Tn,i,u1 and Tn,i,u2 be coorthosimplices in bd(Sn ) lying in the same side of the spherical subspace determined by v1 , . . . , vi−1 , vi+1 , . . . , vn . Suppose that v1 , u∗1 g < v1 , u∗2 g

and

∗∗ v1 , u∗∗ 1 g < v1 , u2 g .

5u2 intersect one another at vi+1 and vi−1 Then, the spherical segments u15 a point in the relative interior of both. Proof. Since both the spherical subspaces S and S , determined by vi−1 , ∗ ∗∗ u∗1 , u∗∗ 1 , and vi+1 , and by vi−1 , u2 , u2 , and vi+1 , respectively, are orthogonal to both spherical simplices v1 v2 . . . vi−1 and vi+1 vi+2 . . . vn , it follows that S = S . Then, the assertion follows from Assertion 7.1 and routine argument (see Figure 7.4). 2 u∗2 u∗1

u∗∗ 1

u∗∗ 2

u2

u1

vi+1

vi−1 vn

v1 Figure 7.4 Let Tn = v1 v2 . . . vn be a spherical simplex in bd(Sn ) with s(Tn ) = 0. Then, we deﬁne s(Ω(v1 , φ) ∩ Tn ) δ(Tn , φ) = . (7.23) s(Tn ) Also, if Tm is a spherical simplex in bd(Sn ) with m < n, for i ≤ m we deﬁne δ(Tm , vi , φ) = vlim δ(Tn , φ). (7.24) →v j i m<j≤n

In this case, letting V be the spherical set obtained by rotating Tm about the spherical subspace determined by v1 , . . . , vi−1 , vi+1 , . . . , vm , by (7.23) and (7.24) it follows that δ(Tm , vi , φ) =

s(Ω(v1 , φ) ∩ V ) . s(V )

7.3. The Coxeter-B¨ or¨ oczky Upper Bound for k(Sn )

117

Now we state and prove the following basic lemma. Lemma 7.5 (B¨ or¨ oczky [1]). Let Tm,i,u1 and Tm,i,u2 be coorthosimplices in bd(Sn ) such that v1 , u∗1 g < v1 , u∗2 g

and

∗∗ v1 , u∗∗ 1 g < v1 , u2 g ,

and let φ < π/4 be a positive number such that Ω(v1 , φ) intersects the spherical subspace Sj◦ in at most one point, where Sj◦ is determined by v2 , . . . , vi−1 , uj , vi+1 , . . . , vm . Then, δ(Tm,i,u1 , u1 , φ) ≥ δ(Tm,i,u2 , u2 , φ). In particular, let T = v1 v2 . . . vm and T = v1 v2 . . . vn be orthosimplices such that v1 , vm g ≥ v1 , vn g

and

v1 , vj g ≥ v1 , vj g

for j = 2, 3, . . . , m − 1, and let φ < π/4 be a positive number such that φ ≤ v1 , v2 g . Then, δ(T, vm , φ) ≤ δ(T , φ). Proof. To prove the ﬁrst part, we apply induction on m. It is obvious for m = 2. Supposing that it is true for m < k ≤ n, we proceed to prove the case m = k. 5u2 , Let p and q be points of the spherical segments u15 vi+1 and vi−1 respectively. For k = n, let v = v(p) be the area of Tn,i,p , and let w = w(q) be the area of Tn,i,q . If k < n, let v(p) and w(q) be the areas of the spherical bodies obtained by rotating Tk,i,p and Tk,i,q , respectively, about the spherical subspace S determined by v1 , . . . , vi−1 , vi+1 , . . . , vk . Then, we write f1 (v) = δ(v1 . . . vi−1 pvi+2 . . . vk , p, φ) and f2 (w) = δ(v1 . . . vi−2 qvi+1 . . . vk , q, φ). By Assertion 7.1 and the inductive assumption it follows that f1 (v) is an increasing function of v, and f2 (w) is a decreasing function of w. Without loss of generality, we may assume that Tk,i,u1 and Tk,i,u2 lie in a common subspace on the same side of the subspace determined by v1 , . . . , vi−1 , vi+1 , . . . , vk . First, we deal with the case 2 < i < k. By 5u2 and u15 Assertion 7.2, the two spherical segments vi−1 vi+1 intersect one another at a point w, say. It is easy to see that % v(u1 ) 1 δ(Tk,i,u1 , u1 , φ) = f1 (v)dv, v(u1 ) 0 δ(Tk,i,u2 , u2 , φ) =

1 w(u2 )

%

w(u2 )

f2 (w)dw, 0

118

7. Upper Bounds for δ(Sn ) and k(Sn ), II

and δ(Tk,i,w , w, φ) =

1 v(w)

1 = w(w)

%

v(w)

f1 (v)dv 0

%

w(w)

f2 (w)dw. 0

Since f1 (v) is increasing and f2 (w) is decreasing, we have δ(Tk,i,u1 , u1 , φ) ≥ δ(Tk,i,w , w, φ) ≥ δ(Tk,i,u2 , u2 , φ). The other cases i = 2 and i = k are routine. Thus, we have proved the ﬁrst part of our lemma. Now we deal with the second part. If m = n, letting Ti = v1 v2 . . . vi vi+1 . . . vn be an orthosimplex such that v1 , vj g = v1 , vj g ,

j = 2, 3, . . . , i,

by the ﬁrst assertion of this lemma it follows that δ(T, φ) ≤ δ(T2 , φ) ≤ . . . ≤ δ(Tn , φ) = δ(T , φ).

(7.25)

When m < n, by choosing vm+1 , vm+2 , . . . , vn near to vm and applying (7.25), it follows that δ(T, vm , φ) ≤ δ(T , φ). 2

Lemma 7.5 is proved.

With these preparations, we can prove the following key lemma of this section. Lemma 7.6 (B¨ or¨ oczky [1]). Let φ < π/4 be a positive number, and let X = {x1 , x2 , . . . , xm(n,φ) } be a discrete set in bd(Sn ) such that the corresponding caps Ω(xi , φ) together form a packing in bd(Sn ). Then, s(Ω(xi , φ)) ≤ σn (φ) s(D (xi )) holds for every point xi ∈ X. Proof. Without loss of generality, we consider Ω(x1 , φ) and D (x1 ) and, for convenience, abbreviate them to Ω and D, respectively. Let Fn−i−1 be an (n − i − 1)-dimensional face of D. By Lemma 7.3 it follows that x1 , Fn−i−1 g ≥ gi (φ).

7.3. The Coxeter-B¨ or¨ oczky Upper Bound for k(Sn )

119

◦ be the Let Ω be the cap {x ∈ bd(Sn ) : x1 , xg < gn−1 (φ)}, let Sn−i−1 ◦ spherical subspace determined by Fn−i−1 , and let pi be the point of Sn−i−1 satisfying ◦ x1 , pi g = min x1 , xg : x ∈ Sn−i−1 .

First, we assert that pi ∈ Fn−i−1 if Ω ∩ Fn−i−1 = ∅. If, on the contrary, pi ∈ Fn−i−1 , we proceed to show Ω ∩ Fn−i−1 = ∅. Assume that Fn−i−1 is a common face of i + 1 spherical Dirichlet-Voronoi cells, D (x1 ), D (x2 ), . . . , D (xi+1 ). There is an (n − i − 2)-dimensional face Fn−i−2 of Fn−i−1 ◦ such that the corresponding subspace Sn−i−2 separates pi from Fn−i−1 (in ◦ Sn−i−1 ). Assume that Fn−i−2 is a common face of D (x1 ), D (x2 ), . . . , D (xn+2 ) and write S = {x ∈ bd(Sn ) : x1 , xg = xi+2 , xg } . Then S separates pi from D, which means that S intersects the spherical segment x5 1 pi . Thus, by Lemma 7.4 it follows that x1 , pi g ≥ g(φ), which implies that Ω ∩ Fn−i−1 = ∅. The ﬁrst assertion follows from this contradiction. Second, we proceed to show s(Ω) ≤ σn (φ). s(D ∩ Ω )

(7.26)

To do this, we decompose D ∩ Ω into smaller cells and consider the local densities associated to them. For convenience, denote the set formed by adjoining the points of conv{v1 , v2 , . . . , vj } to a set Y by [v1 v2 . . . vj Y ], and denote the relative boundary of Ω by bd(Ω ). The decomposition is deﬁned by induction as follows. To start, write v1 = x1 and divide D ∩ Ω into a set [v1 D ∩ bd(Ω )] and sets [v1 Fn−2 ∩ Ω ], where Fn−2 runs over the (n − 2)-dimensional faces of D. Suppose that in the kth step (1 ≤ k < n − 1) we have decomposed D ∩ Ω into sets of type [v1 . . . vj Fn−j ∩ bd(Ω )] for j = 1, 2, . . . , k − 1 (where Fn−1 = D), and sets of type [v1 . . . vk−1 Fn−k ∩ Ω ]. We do the following in the next step: Consider one set [v1 . . . vk−1 Fn−k ∩ Ω ] of the second type ◦ . By the ﬁrst and take vk to be the point pk−1 corresponding to Sn−k assertion it follows that vk ∈ Fn−k ∩ Ω if Fn−i ∩ Ω = ∅. Then, we divide [v1 . . . vk−1 Fn−k ∩ Ω ] into a subset [v1 . . . vk Fn−k ∩ bd(Ω )] and subsets [v1 . . . vk Fn−k−1 ∩ Ω ], where Fn−k−1 runs over all (n − k − 1)dimensional faces of Fn−k . By Lemma 7.3, all vertices of D lie outside of Ω , which means that there is no set of type [v1 . . . vn−1 F0 ∩ Ω ]. Thus, the process ends at the (n − 1)th step. As a result, we have decomposed D ∩ Ω into sets of type [v1 . . . vj Fn−j ∩ bd(Ω )], j = 1, 2, . . . , n − 1 (see Figure 7.5). In particular, the sets of type [v1 . . . vn−1 F0 ∩ bd(Ω )] are orthosimplices.

120

7. Upper Bounds for δ(Sn ) and k(Sn ), II

Fn−k

Ω

vk

Figure 7.5 Third, writing Πj = [v1 . . . vj Fn−j ∩ bd(Ω )], we proceed to show that δ(Πj , φ) ≤ σn (φ).

(7.27)

Let Ω(ui , φ), i = 1, 2, . . . , n, be n mutually touching caps, and let T be the regular spherical simplex spanned by the centers of these caps. T can be divided into n! congruent orthosimplices. One of them, say T = t1 t2 . . . tn , is obtained by taking t1 to be a vertex of T , and when t1 , t2 , . . . , ti−1 have been chosen, taking ti to be the centroid of one of the i-dimensional faces of T that contains t1 , t2 , . . . , ti−1 . Then, we have δ(T, φ) = σn (φ).

(7.28)

Consider the set Πj and let vj+1 be an arbitrary point of Fn−j ∩ bd(Ω ). It is obvious that δ(Πj , φ) ≤ δ(v1 v2 . . . vj+1 , vj+1 , φ).

(7.29)

By Lemma 7.3 it follows that v1 , vi g ≥ gi−1 (φ) for i = 1, 2, . . . , j. Also, it is obvious that v1 , vj+1 g = gn−1 (φ). On the other hand, by deﬁnition, we have t1 , ti g = gi−1 (φ) for i = 2, . . . , n. Thus, (7.29), Lemma 7.5, and (7.28) imply that δ(Πj , φ) ≤ δ(v1 v2 . . . vj+1 , vj+1 , φ) ≤ δ(T, φ) = σn (φ), which proves (7.27). Obviously, (7.27) implies (7.26), and therefore the lemma is proved. 2

7.3. The Coxeter-B¨ or¨ oczky Upper Bound for k(Sn )

121

Proof of Conjecture 7.1. Let φ < π/4 be a positive number and let X = {x1 , x2 , . . . , xm(n,φ) } be a set of points in bd(Sn ) such that the caps Ω(xi , φ), xi ∈ X, form a packing in bd(Sn ). By Lemma 7.6 it follows that

m(n,φ) s(Ω(xi , φ)) m(n, φ)µ(n, φ) = i=1 m(n,φ) nωn s(D (xi )) i=1 s(Ω(xi , φ)) ≤ max 1≤i≤m(n,φ) s(D (xi )) ≤ σn (φ). Let T = v1 v2 . . . vn be a regular spherical simplex in bd(Sn ) of geodesic side length 2φ and corresponding dihedral angle 2α. Thus, m(n, φ) ≤

n2 ωn s(T ∩ Ω(v1 , φ)) nωn σn (φ) = . µ(n, φ) s(T )s(Ω(v1 , φ))

(7.30)

From the deﬁnition of Schl¨ aﬂi’s function, we have s(T ) =

n!nωn Fn (α). 2n

(7.31)

Also, by projecting T and Ω(v1 , φ) onto the tangent hyperplane of Sn at v1 , it is easy to see that s(T ∩ Ω(v1 , φ)) (n − 1)! = Fn−1 (α). s(Ω(v1 , φ)) 2n−1

(7.32)

Then, (7.30), (7.31), and (7.32) together yield m(n, φ) ≤

nωn σn (φ) 2Fn−1 (α) = . µ(n, φ) Fn (α)

(7.33)

In order to determine the relationship between φ and α, we consider a special simplex T . Let b and d be suitable numbers such that T is determined by vertices v1 = (b + d, b, . . . , b), v2 = (b, b + d, b, . . . , b), . . . , vn = (b, b, . . . , b, b + d). Then, it follows from elementary geometry that cos 2φ =

v1 , v2 2bd + nb2 = v1 2 d2 + 2bd + nb2

and therefore

d2 + 1. (7.34) 2bd + nb2 On the other hand, the internal angle 2α between two of the n bounding hyperplanes n Hi = x : ui , x = (nb + d)xi − b xj = 0 , sec 2φ =

j=1

122

7. Upper Bounds for δ(Sn ) and k(Sn ), II

where ui are suitable vectors, is given by cos 2α =

u1 , u2 2bd + nb2 = 2 . 2 u1 d + (n − 1)(2bd + nb2 )

Then, by (7.34) we have sec 2α =

d2 + n − 1 = sec 2φ + n − 2. 2bd + nb2

Conjecture 7.1 follows from (7.33) and (7.35).

(7.35) 2

Proof of Theorem 7.2. Using (7.19), (7.18), and Lemma 7.2, since sec 2α = n when φ = π/6, it follows that √ 2Fn−1 (α) πn3 k(Sn ) = m(n, π/6) ≤ ∼ √ 20.5n . Fn (α) e 2 Theorem 7.2 is proved.

2

7.4. Claude Ambrose Rogers Claude Ambrose Rogers was born in Cambridge in 1920, but his family soon moved to north London. He went to boarding school in Berkhamsted, Herts, and then in 1938 he studied mathematics at University College London, being evacuated to Bangor for the ﬁrst year of the war. His Ph.D thesis on divergent series was written under the supervision of R.G. Cooke of Birkbeck College and L.S. Bosanquet of University College London and accepted in 1949. In 1946 he joined the mathematical faculty of University College London. During that time, inﬂuenced by H. Davenport, he became interested in packing and covering, to which he has made contributions of fundamental importance. In 1954 Rogers became the Mason professor of pure mathematics at the University of Birmingham. Four years later, when Davenport left London for Cambridge, Professor Rogers succeeded him as the Astor professor of pure mathematics at University College London, where he worked until his retirement in 1986. Besides Theorem 3.2, Theorem 3.3, Theorem 3.4, and Theorem 7.1, Professor Rogers has made important contributions to the geometry of numbers, convex geometry, and geometric measure theory. For example, in 1957 in a joint paper he and G.C. Shephard proved that for every ndimensional convex body K,

v(D(K)) 2n 2n ≤ ≤ , v(K) n

7.4. Claude Ambrose Rogers

123

where the upper bound can be realized if and only if K is a simplex and the lower bound can be realized if and only if K is centrally symmetric. In 1975 in another joint paper he and D.G. Larman proved that when n ≥ 12, there exist n-dimensional centrally symmetric convex bodies C1 and C2 centered at o such that v(C1 ) > v(C2 ) and s(C1 ∩ H) < s(C2 ∩ H) for every hyperplane H that contains o. The ﬁrst example completely solves a problem of W. Blaschke, and the second solves a problem of H. Busemann and C.M. Petty in the corresponding dimensions. He is also the author of the classic book Packing and Covering. Professor Rogers has been a fellow of the Royal Society since 1959, and he was the president of the London Mathematical Society for the term 1970– 1972. For his distinguished contributions, he was awarded a De Morgan medal by the London Mathematical Society in 1977.

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8. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres III

8.1. Jacobi Polynomials The Jacobi polynomials are a family of well-known special functions. They are deﬁned, for k = 0, 1, 2, . . . , as

k 1 k+α k+β α,β (t + 1)i (t − 1)k−i , Pk (t) = k 2 i=0 i k−i where α > −1 and β > −1 are parameters. These polynomials play an important role in obtaining the Kabatjanski-Leven˘stein upper bounds for δ(Sn ) and k(Sn ). In this section we introduce some of their fundamental properties. Since these properties are well-known and easy to deduce (see Szeg¨ o [1]), we omit their proofs.

Assertion 8.1. Pkα,β (1) =

k+α . k

Assertion 8.2. % 1 Pkα,β (t)Plα,β (t)(1 − t)α (1 + t)β dt = δkl (k, α, β), −1

where δkl is the Kronecker symbol and (k, α, β) =

2α+β+1 Γ(k + α + 1)Γ(k + β + 1) . k!(2k + α + β + 1)Γ(k + α + β + 1)

126

8. Upper Bounds for δ(Sn ) and k(Sn ), III

Conversely, if P0 (t), P1 (t), P2 (t), . . . is a sequence of polynomials such that %

1

−1

Pk (t)Pl (t)(1 − t)α (1 + t)β dt = δkl (k, α, β),

where the degree of Pk (t) is k, then Pk (t) = ±Pkα,β (t),

k = 0, 1, 2, . . . .

Assertion 8.3. The Jacobi polynomials y = Pkα,β (t) satisfy the diﬀerential equation (1 − t2 )y + [β − α − (α + β + 2)t]y + k(k + α + β + 1)y = 0. Assertion 8.4 (The Christoﬀel-Darboux Formula). α,β α,β Pk+1 (t)Pkα,β (s) − Pkα,β (t)Pk+1 (s) t−s

(2k + α + β + 2)(2k + α + β + 1) (k, α, β) α,β Pi (t)Piα,β (s). 2(k + 1)(k + α + β + 1) (i, α, β) i=0 k

=

Clearly, by taking s → t one obtains the following result. Corollary 8.1.

α,β α,β (t) Pkα,β (t) − Pk+1 (t) Pkα,β (t) > 0. Pk+1

Assertion 8.5. Let t1 (k, α, β), t2 (k, α, β), . . . , tk (k, α, β) be the k zeros (in decreasing order) of the polynomial Pkα,β (t). Then, for k = 1, 2, . . . and i = 1, 2, . . . , k, ti+1 (k + 1, α, β) < ti (k, α, β) < ti (k + 1, α, β) < 1. Assertion 8.6. When α = β = (n − 3)/2, Pkα,β (t)Plα,β (t) =

k+l

ai (k, l)Piα,β (t)

i=0

holds for some suitable nonnegative coeﬃcients ai (k, l).

8.2. Delsarte’s Lemma

127

8.2. Delsarte’s Lemma A spherical code ℵ of dimension n, cardinality m, and minimum angle ϕ is a set of m points of bd(Sn ) such that x, y ≤ cos ϕ holds for any two distinct points x and y of ℵ. In other words, the m caps Ω(x, ϕ/2), x ∈ ℵ, form a packing in bd(Sn ). For convenience, we call ℵ an {n, m, ϕ} code. Then, we deﬁne m[n, ϕ] to be the maximum size, m, of any such spherical code. Clearly, m(n, ϕ/2) = m[n, ϕ].

(8.1)

As an upper bound for m[n, ϕ], we have the following remarkable result. Lemma 8.1 (Delsarte [1]). Write α = (n − 3)/2. Let f (t) =

k

fi Piα,α (t)

i=0

be a real polynomial such that f0 > 0, fi ≥ 0 for i = 1, 2, . . . , k, and f (t) ≤ 0 for −1 ≤ t ≤ cos ϕ. Then m[n, ϕ] ≤

f (1) . f0

Let L be the linear space of square integrable real functions over bd(Sn ). For two functions f1 (x) and f2 (x) of L we deﬁne the inner product % 1 f1 ◦ f2 = f1 (x)f2 (x)dω(x), nωn bd(Sn ) where ω(x) indicates the surface element of bd(Sn ) at x. Let Pk be the subspace of L consisting of spherical polynomials of degree at most k, and let Hk be the subspace of L consisting of homogeneous harmonic polynomials of degree k. Here, we say f (x) is a harmonic if ∆f =

∂2f ∂2f ∂2f + + · · · + = 0, ∂x21 ∂x22 ∂x2n

where ∆ is the Laplace operator. Assertion 8.7. Pk = Hk ⊕ Hk−1 ⊕ · · · ⊕ H0 . Proof. A polynomial pk (x) in Pk is a linear combination of monomials m(x) = xk11 xk22 · · · xknn

128

8. Upper Bounds for δ(Sn ) and k(Sn ), III

with k1 + k2 + · · · + kn = l ≤ k. Since x ∈ bd(Sn ), we have x, x = 1 and therefore m(x) = x, xi m(x). Take i = (k − l)/2. Then x, xi m(x) is a homogeneous polynomial, of degree k if k − l is even and of degree k − 1 if k − l is odd. Denote the subspace of L consisting of homogeneous polynomials of degree k by Hk∗ . Then pk (x) = h∗k (x) + h∗k−1 (x), ∗ where h∗k (x) ∈ Hk∗ and h∗k−1 (x) ∈ Hk−1 . Also, since

h∗k ◦ h∗k−1 = (−1)2k−1 h∗k ◦ h∗k−1 , we have

h∗k ◦ h∗k−1 = 0.

Thus,

∗ . Pk = Hk∗ ⊕ Hk−1

(8.2)

By routine argument it is easy to see that ∆ is a linear operator on Hk∗ ∗ with kernel Hk and image Hk−2 . On the other hand, it is well-known from real analysis (see Groemer [5]) that % % f (x)∆g(x)dω(x) = g(x)∆f (x)dω(x) bd(Sn )

bd(Sn )

holds for every pair of twice continuously diﬀerentiable functions f (x) and ∗ , we g(x) on bd(Sn ). In particular, for hk (x) ∈ Hk and h∗k−2 (x) ∈ Hk−2 have % 1 hk ◦ h∗k−2 = g ∗ (x)∆hk (x)dω(x) = 0, nωn bd(Sn ) k where gk∗ (x) is a suitable function satisfying h∗k−2 (x) = ∆gk∗ (x). Thus, we have

∗ Hk∗ = Hk ⊕ Hk−2 .

(8.3)

Applying (8.2) and (8.3) inductively, it follows that Pk = Hk ⊕ Hk−1 ⊕ · · · ⊕ H0 . Assertion 8.7 is proved.

2

Next we introduce a special class of spherical harmonics, which play an important role in our considerations. Since Hk is a ﬁnite-dimensional

8.2. Delsarte’s Lemma

129

linear space equipped with a nontrivial inner product, it follows from a fundamental result of functional analysis that for any real-valued linear functional F (hk ) deﬁned on Hk there is a unique g ∈ Hk such that F (hk ) = hk ◦ g. In particular, for each point x ∈ bd(Sn ), by taking F (hk ) = hk (x) it is easy to see that there is a unique function Qk (x, y) ∈ Hk such that % 1 hk (x) = hk (y)Qk (x, y)dω(y). (8.4) nωn bd(Sn ) Usually, Qk (x, y) is called a zonal spherical harmonic. Assertion 8.8. For every isometry σ of En that leaves the origin ﬁxed, Qk (σx, σy) = Qk (x, y). Proof. Assume that x = σy = yA, where A is an n × n matrix satisfying AA = I, and assume hk (x) ∈ Hk . Then, routine computation yields n ∂ 2 hk ∆hk (σy) = aik ajk ∂xi ∂xj i, j k=1

= ∆hk (x) = 0.

(8.5)

This means that hk (σx) ∈ Hk if hk (x) ∈ Hk . In particular, Qk (σx, σy), as a function of y, is a spherical harmonic. Write % 1 Jσ = hk (y)Qk (σx, σy)dω(y). nωn bd(Sn ) By substituting z = σy and using (8.4), we have % 1 Jσ = hk (σ−1 z)Qk (σx, z)dω(z) nωn bd(Sn ) = hk (σ−1 σx) = hk (x) % 1 = hk (y)Qk (x, y)dω(y). nωn bd(Sn ) Then, Assertion 8.8 follows from the uniqueness of Qk (x, y).

2

Assertion 8.9 (Addition Formula). Let hk,1 (x), hk,2 (x), . . . , hk,d(k) (x) be an orthonormal basis of Hk , where d(k) is the dimension of Hk . Then,

d(k)

Qk (x, y) =

i=1

hk,i (x)hk,i (y).

130

8. Upper Bounds for δ(Sn ) and k(Sn ), III

Proof. Any function hk (x) ∈ Hk can be expanded as

d(k)

hk (x) =

hk ◦ hk,i hk,i (x).

i=1

In particular, using (8.4),

d(k)

Qk (x, y) =

Qk (x, y) ◦ hk,i (y)hk,i (y)

i=1

d(k)

=

hk,i (x)hk,i (y).

i=1

2

Assertion 8.9 is proved.

By Assertion 8.8 it follows that the value of Qk (x, y) is determined by k and x, y. So, for convenience, we write Gk (x, y) = Qk (x, y). Assertion 8.10. Write α = (n − 3)/2. Then Gk (t) = ck Pkα,α (t) holds for some positive number ck . Proof. By Assertion 8.7, for k = l, we have % J= Qk (x, y)Ql (x, y)dω(y) = 0. bd(Sn )

Then, substituting t = x, y and dω(y) = (n − 1)ωn−1 (1 − t2 )(n−3)/2 dt, we obtain % J = (n − 1)ωn−1

1

−1

Gk (t)Gl (t)(1 − t2 )(n−3)/2 dt = 0.

Thus, by the second part of Assertion 8.2 we have Gk = ck Pkα,α (t),

k = 0, 1, . . . .

To determine the sign of ck , we observe that % 1 Gk (1) = Gk (1)dω(x) nωn bd(Sn )

(8.6)

8.2. Delsarte’s Lemma

131 1 = nωn

%

d(k)

h2k,i (x)dω(x)

bd(Sn ) i=1

1 % h2k,i (x)dω(x) nω n bd(S ) n i=1

d(k)

=

d(k)

=

1 = d(k).

(8.7)

i=1

Then, it follows from (8.6), (8.7), and Assertion 8.1 that ck > 0 holds for k = 0, 1, . . . . Assertion 8.10 is proved. 2 We can now proceed to prove Lemma 8.1. Proof of Lemma 8.1. Let X be a discrete subset of bd(Sn ) such that card{X} = m[n, ϕ], and Ω(x, ϕ/2), x ∈ X, form a cap packing in bd(Sn ). Then, x, y ≤ cos ϕ holds for any two distinct points x and y of X. Let k f (t) = fi Piα,α (t)

(8.8)

i=0

be a real polynomial such that f0 > 0, fi ≥ 0 (i = 1, 2, . . . , k), and f (t) ≤ 0 for −1 ≤ t ≤ cos ϕ. For convenience, we apply f (t) =

k

gi Gi (t)

i=0

rather than (8.8), where g0 = f0 , G0 (t) = 1, and gi = fi /ci ≥ 0 for i = 1, 2, . . . , k. Write J= f (x, y). x, y∈X

Let W be the set of numbers w such that w = x, y holds for some pair of distinct points x and y of X. For w ∈ W let p(w) be the number of ordered pairs (x, y), x, y ∈ X, such that x, y = w. Then, we have J = m[n, ϕ]f (1) +

w∈W

p(w)f (w).

(8.9)

132

8. Upper Bounds for δ(Sn ) and k(Sn ), III

On the other hand, since G0 (w) = Q0 (x, y) = 1, the addition formula implies that J =

k i=0

gi

k

Gi (x, y) =

i=0

x, y∈X

= g0 m[n, ϕ]2 +

2

= f0 m[n, ϕ] +

k

gi

x, y∈X

k

d(i)

gi

j=1

Qi (x, y)

x, y∈X

Qi (x, y)

i=1

i=1

gi

2 hi,j (x) .

(8.10)

x∈X

Comparing (8.9) and (8.10), since f (w) ≤ 0 for w ∈ W , we have f (1) m[n, ϕ] ≥ f0 m[n, ϕ]2 and therefore m[n, ϕ] ≤

f (1) . f0 2

This proves Lemma 8.1.

Remark 8.1. This fundamental lemma was ﬁrst proved by Delsarte in [1] and [2]. Alternative proofs can be found in Delsarte, Goethals, and Seidel [1], Kabatjanski and Leven˘ stein [1], Odlyzko and Sloane [1], Lloyd [1], and Seidel [1]. Our proof follows Seidel’s presentation.

8.3. The Kabatjanski-Leven˘stein Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres For convenience, we write α = (n − 3)/2 and denote by F(n, ϕ) the family of polynomials k f (t) = fi Piα,α (t), i=0

such that f0 > 0, fi ≥ 0 for i = 1, 2, . . . , k, and f (t) ≤ 0 for −1 ≤ t ≤ cos ϕ. Then, we deﬁne f (1) m∗ [n, ϕ] = inf . f ∈F (n,ϕ) f0 It is obvious that

m[n, ϕ] ≤ m∗ [n, ϕ].

(8.11)

8.3. The Kabatjanski-Leven˘ stein Upper Bounds

133

As an upper bound for m[n, ϕ], we have the following fundamental result. Lemma 8.2 (Kabatjanski and Leven˘stein [1]). and n → ∞,

When 0 < ϕ < π/2

log m[n, ϕ] 1 + sin ϕ 1 + sin ϕ 1 − sin ϕ 1 − sin ϕ log − log . n 2 sin ϕ 2 sin ϕ 2 sin ϕ 2 sin ϕ The proof of this deep lemma requires some careful preparation. Assertion 8.11. Write s = cos ϕ and τ =−

α,α k + 1 Pk+1 (s) . k + α + 1 Pkα,α (s)

When t1 (k, α, α) < s < t1 (k + 1, α, α) (see Assertion 8.5), m∗ [n, ϕ] ≤

k+2α+1 (1 + τ )2 k . (1 − s)τ

Proof. Consider the polynomial (k + α + 1)(2k + 2α + 1) α,α α,α f (t) = Pk+1 (t)Pkα,α (s) − Pkα,α (t)Pk+1 (s) (k + 1)(k + 2α + 1) ×

k (k, α, α) i=0

(i, α, α)

Piα,α (s)Piα,α (t)

(8.12)

of degree 2k + 1. By Assertion 8.5 and our assumption it follows that α,α Pkα,α (s)Piα,α (s) > 0 and Pk+1 (s)Piα,α (s) < 0

for all i ≤ k. Hence, by Assertions 8.1 and 8.6 we have f0 > 0 and fi ≥ 0 for i = 1, 2, . . . , 2k + 1. Also, by Assertion 8.4, we have f (t) =

2 α,α α,α Pk+1 (t)Pkα,α (s) − Pkα,α (t)Pk+1 (s)

(8.13)

t−s

and therefore f (t) ≤ 0 when −1 ≤ t ≤ s. Thus, f (t) ∈ F(n, ϕ). By Assertion 8.1 and (8.13) it follows that 2

f (1) =

(Pkα,α (s)) 1−s

k+α k

2

α,α k + α + 1 Pk+1 (s) − α,α k+1 Pk (s)

2 .

134

8. Upper Bounds for δ(Sn ) and k(Sn ), III

On the other hand, (8.12) and the ﬁrst part of Assertion 8.2 together yield % f0 (0, α, α) =

1

−1

f (t)(1 − t2 )α dt

α,α = − Pk+1 (s)Pkα,α (s)

(k + α + 1)(2k + 2α + 1)(k, α, α) . (k + 1)(k + 2α + 1)

Hence α,α f0 = −Pk+1 (s)Pkα,α (s)

(k + α + 1)(2k + 2α + 1)(k, α, α) . (k + 1)(k + 2α + 1)(0, α, α)

Then, by Lemma 8.1, the deﬁnition of (i, α, α), and routine computation we obtain k+2α+1 (1 + τ )2 f (1) ∗ k m [n, ϕ] ≤ ≤ . f0 (1 − s)τ 2

Assertion 8.11 is proved. Assertion 8.12. When s = cos ϕ ≤ t1 (k, α, α), 4 k+2α+1 ∗ k m [n, ϕ] ≤ . 1 − t1 (k + 1, α, α)

Proof. Regard τ as a function of s. When s decreases from t1 (k + 1, α, α) to t1 (k, α, α), it follows from Corollary 8.1 that τ increases from 0 to ∞. So, τ = 1 holds at a certain s = cos ϕ with t1 (k, α, α) < s < t1 (k + 1, α, α). Since m∗ [n, ϕ] is an increasing function of s, it follows from Assertion 8.11 that 4 k+2α+1 ∗ ∗ k m [n, ϕ] ≤ m [n, ϕ ] ≤ 1 − s k+2α+1 4 k . ≤ 1 − t1 (k + 1, α, α) 2

Assertion 8.12 is proved. Assertion 8.13. If c is a positive number such that lim

k→∞

then

α n−3 1 = lim = , k→∞ k 2k 2c

c(1 + c) lim t1 (k, α, α) = . k→∞ 1 + 2c 2

(8.14)

8.3. The Kabatjanski-Leven˘ stein Upper Bounds

135

Proof. Consider the function y = (1 − t2 )(α+1)/2 Pkα,α (t),

(8.15)

which has the same zeros in −1 < t < 1 as Pkα,α (t). From Assertion 8.3, it can be deduced that this function satisﬁes the diﬀerential equation y + g(t)y = 0, where g(t) =

(8.16)

η(γ 2 − t2 ) , (1 − t2 )2

η = (k + α)(k + α + 1), and γ=

[k(k + 2α + 1) + α + 1]/η.

Since γ < 1, g(t) is negative in γ < t < 1, and y → 0 when t → 1. Thus, by (8.16) it follows that y cannot vanish for γ ≤ t < 1. Consequently, t1 (k, α, α) < γ.

(8.17)

On the other hand, for any positive number b the equation y + b2 y = 0 has a solution y = sin(bt + d), where d is a constant, that has a zero between t1 = γ−2π/b and t2 = γ−π/b. If we choose b such that t1 ≥ −1 and g(t) ≥ b2 for t1 ≤ t ≤ t2 , the function (8.15) will have a zero in this interval, and hence t1 (k, α, α) ≥ γ − In fact, taking b=

2π . b

πγη 1/4

, 4 it can be veriﬁed that t1 ≥ −1, γ ≥ 2π/b, and g(t) ≥

(4πγb − 4π 2 )η b2 ≥ b 2 [b2 − (γb)2 + 2πγb − π 2 ]2

holds for t1 ≤ t ≤ t2 when k is suﬃciently large. Then, by (8.17), (8.18), and routine computation we obtain 2 c(1 + c) lim t1 (k, α, α) = γ = . k→∞ 1 + 2c

(8.18)

136

8. Upper Bounds for δ(Sn ) and k(Sn ), III 2

Assertion 8.13 is proved. Proof of Lemma 8.2. Take c=

1 − sin ϕ 2 sin ϕ

(8.19)

and apply Assertion 8.13. This gives lim t1 (k, α, α) = cos ϕ.

k→∞

Then, by (8.11) and Assertion 8.12 it follows that 4 k+2α+1 k m[n, ϕ] ≤ 1 − t1 (k + 1, α, α)

k + 2α + 1 4 . 1 − cos ϕ k

(8.20)

Applying (8.14), Stirling’s formula, and (8.19) to (8.20) we obtain log m[n, ϕ] (1 + c) log(1 + c) − c log c n 1 + sin ϕ 1 + sin ϕ 1 − sin ϕ 1 − sin ϕ = log − log . 2 sin ϕ 2 sin ϕ 2 sin ϕ 2 sin ϕ 2

Lemma 8.2 is proved.

The special case ϕ = π/3 of Lemma 8.2 yields the following upper bound for the kissing numbers of spheres. Theorem 8.1 (Kabatjanski and Leven˘stein [1]). k(Sn ) ≤ 20.401n(1+o(1)). To get a corresponding upper bound for the packing densities of spheres, we need another lemma. Lemma 8.3. δ(Sn ) ≤

sin

ϕ n m[n + 1, ϕ] 2

holds for a certain small ϕ. Proof. Let r be a large number and let H be a hyperplane through the center of rSn+1 . In H we construct an n-dimensional packing of unit spheres with density δ(Sn ). By shifting the packing slightly we may assume that the portion of H inside rSn+1 contains at least rn δ(Sn ) centers. We project these centers, perpendicularly to H, “upwards” onto bd(rSn+1 ). The Euclidean distance between the new points is at least 2, and their geodesic distance is at least ϕ, where sin

ϕ 1 = . 2 r

8.3. The Kabatjanski-Leven˘ stein Upper Bounds

137

Thus, we have rn δ(Sn ) ≤ m[n + 1, ϕ] and therefore δ(Sn ) ≤

sin

ϕ n m[n + 1, ϕ]. 2 2

Lemma 8.3 is proved.

Remark 8.2. This lemma ﬁrst appeared in the appendix of the Russian edition of L. Fejes T´ oth [9] (translated by I.M. Yaglom). Here, we follow the proof of Sloane [4]. Theorem 8.2 (Kabatjanski and Leven˘stein [1]). δ(Sn ) ≤ 2−0.599n(1+o(1)). Proof. Routine computation yields 1 + sin ϕ 1 + sin ϕ 1 − sin ϕ 1 − sin ϕ log − log 2 sin ϕ 2 sin ϕ 2 sin ϕ 2 sin ϕ 1 ≤ − log(1 − cos ϕ) − 0.099 2 when 0 < ϕ ≤ ϕ0 ≈ 63◦ . Then, it follows from Lemma 8.2 that m[n, ϕ] ≤ (1 − cos ϕ)−0.5n 2−0.099n ϕ −n −0.599n = sin 2 . 2 Consequently, by Lemma 8.3 we have ϕ n ϕ −(n+1) −0.599(n+1) 2 δ(Sn ) ≤ sin sin 2 2 ϕ −1 −0.599n(1+o(1)) = sin 2 2 for some suitable ϕ. Theorem 8.2 follows.

(8.21)

2

To end this chapter let us propose two open problems. Problem 8.1. What is the behavior of f (n) = k(Sn ) − k∗ (Sn )? Does f (n) = 0 (or = 0) hold for inﬁnitely many dimensions? What is the asymptotic order of lim f (n)? Problem 8.2. What is the behavior of g(n) =

δ(Sn ) ? δ ∗ (Sn )

Does g(n) = 1 (or = 1) hold for inﬁnitely many dimensions? What is the asymptotic order of lim g(n)?

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9. The Kissing Numbers of Spheres in Eight and Twenty–Four Dimensions

9.1. Some Special Lattices Let Λ be an n-dimensional lattice. As usual, we call it an integral lattice if u, u ∈ Z,

u ∈ Λ.

In particular, we call it an even integral lattice if u, u ∈ 2Z,

u ∈ Λ,

and an odd integral lattice otherwise. There are some special integral lattices that play important roles in diﬀerent areas of mathematics. For example, Zn = z ∈ E n : z i ∈ Z , n+1 An = z ∈ Zn+1 : zi = 0 , i=1

n Dn = z ∈ Zn : zi ∈ 2Z , i=1

8 1 E8 = u ∈ Z8 : ui − uj ∈ Z, ui ∈ 2Z , 2 i=1 8 E7 = u ∈ E8 : ui = 0 , i=1

140

9. The Kissing Numbers of S8 and S24

E6 =

6 u ∈ E8 : ui = u 7 + u 8 = 0 , i=1

and the√well-known √ Leech lattice Λ24 , which is determined √ by the basis √ a1 = 2√ 2e1 , a2 = 2(e1 √ + e2 ), a3 = 2(e1 + e ), a = 2(e1 + e4 ), 4 √ 3 + e ), a = 2(e + e ), a = 2(e + e ), a a5 = 2(e 5 √6 1 6 √ 7 1 7 √ 8 = (e1 + √1 · · · + e8 )/ 2, a9 = 2(e1 + e9 ), a10 = √ 2(e1 + e10 ),√a11 = 2(e1 + e11 ), a12 = (e1 + · · · + e4 + e9 + · · · + e12 )/√ 2, a13 = 2(e1 + e13 ), a14 = (e1 + e2 + e5 + e6 √ + e9 + e10 + e13 + e14 )/ 2, a15 = (e1 + e3 + e5 + e7 + e√ 9+ e11 + e√ + e )/ 2, a = (e + e + e + e + e + e + e + e )/ 13 15 16 1 4 5 8 9 12 13 16 √2, a17 = 2(e1 + e17 ), a18 = (e1 + e3 + e5 + e8√+ e9 + e10 + e17 + e18 )/ 2, a19 = (e1 +e4 +e5 +e 7+ √6 +e9 +e11 +e17 +e19 )/ 2, a20 = (e1 +e2 +e5 +e√ e9 + e12 + e17 + e20 )/ 2, a21 = (e2 + e3 + e4 + e5 + e9 √ + e13 + e17 + e21 )/ 2, a22 = (e9 + e10 + e13 + e14 + e17 +√ e18 + e21 + e22 )/ 2, a23 = (e9 + e11 √+ e13 + e15 + e17 + e19 + e21 + e23 )/ 2, a24 = (−3e1 + e2 + · · · + e24 )/ 8, where e1 , e2 , . . . , e24 is an orthonormal basis of E 24 . It can be veriﬁed that An (n ≥ 1), Dn (n ≥ 4), En (n = 6, 7, 8), and Λ24 are even integral lattices, and Zn (n ≥ 1) are odd integral lattices. Writing r = min

1

2 u

: u ∈ Λ \ {o}

and M (Λ) = {u ∈ Λ : u = 2r}, it is easy to see that rSn + Λ is a packing and rSn + u touches rSn at its boundary if u ∈ M (Λ). Thus, k∗ (Sn ) = max {card{M (Λ)}} ,

(9.1)

where the maximum is over all n-dimensional lattices. In particular, routine computation yields card {M (E8 )} = 240 (9.2) and card {M (Λ24 )} = 196560.

(9.3)

To end this section we cite a well-known result about the representations of integral lattices that will be useful in Section 4 of this chapter. We omit its extremely complicated proof. Lemma 9.1 √ (Kneser [1]). Every integral lattice generated by vectors of norm 1 and 2 can be written as a direct sum of the special lattices Zn (n ≥ 1), An (n ≥ 1), Dn (n ≥ 4), and En (n = 6, 7, 8).

9.2. Two Theorems of Leven˘stein, Odlyzko, and Sloane

141

9.2. Two Theorems of Leven˘stein, Odlyzko, and Sloane Theorem 9.1 (Leven˘stein [2], Odlyzko and Sloane [1]). k ∗ (S8 ) = k(S8 ) = 240. Proof. Abbreviate the Jacobi polynomial Pk2.5,2.5 (t) to Pk , and deﬁne 320 1 2 2 1 (t + 1) t + t t− 3 2 2 16 200 832 = P0 + P1 + P2 + P3 7 63 231 1216 5120 2560 + P4 + P5 + P6 . 429 3003 4641

f (t) =

Clearly, f (t) satisﬁes the conditions of Lemma 8.1 for ϕ = π/3. Thus, by (8.1) and Lemma 8.1 it follows that k(S8 ) = m(8, π/6) ≤

f (1) = 240. f0

(9.4)

Then, (9.1), (9.2), and (9.4) together yield k ∗ (S8 ) = k(S8 ) = 240. 2

Theorem 9.1 is proved. Theorem 9.2 (Leven˘stein [2], Odlyzko and Sloane [1]). k∗ (S24 ) = k(S24 ) = 196560. Proof. Abbreviate the Jacobi polynomial Pk10.5,10.5 (t) to Pk , and deﬁne 1490944 1 2 1 2 2 1 2 1 (t + 1) t + t+ t t− t− 15 2 4 4 2 48 1144 12992 73888 = P0 + P1 + P2 + P3 + P4 23 425 3825 22185 2169856 59062016 4472832 + P5 + P6 + P7 687735 25365285 2753575 23855104 7340032 7340032 + P8 + P9 + P10 . 28956015 20376455 80848515

f (t) =

Then, f (t) satisﬁes the conditions of Lemma 8.1 for ϕ = π/3. Thus, by (8.1) and Lemma 8.1 we have k(S24 ) = m(24, π/6) ≤

f (1) = 196560. f0

(9.5)

142

9. The Kissing Numbers of S8 and S24

Then, (9.1), (9.3), and (9.5) together yield k ∗ (S24 ) = k(S24 ) = 196560. 2

Theorem 9.2 is proved. Remark 9.1. By constructing a suitable polynomial one can prove that k(S4 ) ≤ 25. On the other hand, by Theorem 2.6, we know that k∗ (S4 ) = 24.

Thus, k(S4 ) is either 24 or 25. However, the correct answer is still unknown. Remark 9.2. By choosing suitable polynomials, numerical upper bounds for δ(Sn ) and k(Sn ) can be obtained in special dimensions. For results of this kind we refer to Conway and Sloane [1].

9.3. Two Principles of Linear Programming For convenience, for two points u and v of En we write u v if ui ≤ vi holds simultaneously for i = 1, 2, . . . , n. Usually, a linear programming problem is to determine p = max {a, x : xA b, o x} ,

(9.6)

where a ∈ E n , b ∈ E m , and A is a m × n matrix. Its dual is to determine q = min {b, y : a yA, o1 y} ,

(9.7)

where o1 is the origin of E m . Now we introduce two fundamental principles concerning these problems that will be useful in the next section. Since they are well-known and can be found in any book on linear programming (see Gr¨ otschel, Lov´ asz, and Schrijver [1] or Nemhauser and Wolsey [1]), we omit their proofs here. Principle 9.1. If p or q is ﬁnite, then p = q. Principle 9.2. If x∗ is an optimal solution of (9.6) and y∗ is an optimal solution of (9.7), writing s∗ = b − x∗ A and t∗ = y∗ A − a, then x∗i t∗i = 0,

i = 1, 2, . . . , n,

9.4. Two Theorems of Bannai and Sloane and

yj∗ s∗j = 0,

143

j = 1, 2, . . . , m.

Remark 9.3. In fact, Principle 9.2 can be easily deduced from Principle 9.1.

9.4. Two Theorems of Bannai and Sloane Let ℵ be an {n, m, ϕ} code. For u ∈ ℵ the inner product distribution of ℵ with respect to u is the set of numbers {a(u, t) : −1 ≤ t ≤ 1}, where a(u, t) = card {v ∈ ℵ : u, v = t} , and the inner product distribution of ℵ is the set of numbers {a(t) : −1 ≤ t ≤ 1}, where 1 a(t) = a(u, t). m u∈ℵ

In particular, writing γ = cos ϕ, we have a(1) = 1, a(t) = 0 for γ < t < 1, and a(t). m−1= −1≤t≤γ

Also, by Assertions 8.10 and 8.9, when α = (n − 3)/2 we have

a(t)Pkα,α (t) =

−1≤t≤1

1 Qk (x, y) mck x, y∈ℵ

=

d(k) 1 hk,i (x)hk,i (y) mck i=1 x, y∈ℵ

d(k)

1 = mck i=1

2 hk,i (x) ≥ 0.

(9.8)

x∈ℵ

Thus, for ﬁxed n and ϕ an upper bound to m is given by the following linear programming problem: p(n, ϕ) = max 1 + a(t) : − a(t)Pkα,α (t) ≤ Pkα,α (1) −1≤t≤γ

−1≤t≤γ

for k ≥ 1, a(t) ≥ 0 .

(9.9)

144

9. The Kissing Numbers of S8 and S24

Its dual can be formulated as l f (1) q(n, ϕ) = min : f (t) = fk Pkα,α (t), f0 > 0, fk ≥ 0 for k ≥ 1; f0 k=0

f (t) ≤ 0 for − 1 ≤ t ≤ γ .

(9.10)

Remark 9.4. Formulae (9.9), (9.10), and Principle 9.1 together imply Lemma 8.1. Theorem 9.3 (Bannai and Sloane [1]). There is a unique way (up to isometry) of arranging 240 nonoverlapping unit spheres in E8 so that they all touch another unit sphere. Proof. Let ℵ∗ = {u∗1 , u∗2 , . . . , u∗240 } be an {8, 240, π/3} code, and let {a∗ (t) : −1 ≤ t ≤ 1} be its inner product distribution. Then {a∗ (t) : −1 ≤ t ≤ 1} is an optimal solution to the linear programming problem (9.9) for n = 8 and ϕ = π/3, and the polynomial f (t) deﬁned in the proof of Theorem 9.1 is an optimal solution to its dual problem (9.10). Since the dual variables f1 , f2 , . . . , f6 are nonzero, by Principle 9.2, (9.8) holds with equality for k = 1, 2, . . . , 6. On the other hand, since the optimal polynomial f (t) does not vanish, except for t = −1, − 12 , 0, and 1 ∗ 2 , the primal variables must vanish everywhere except perhaps for a (−1), 1 ∗ ∗ ∗ 1 a (− 2 ), a (0), and a ( 2 ). From (9.8), these numbers satisfy the equations a∗ (t)Pkα,α (t) = −Pkα,α (1) −1≤t<1

for k = 1, 2, . . . , 6. Solving these equations we obtain a∗ (−1) = 1,

a∗ (− 12 ) = a∗ ( 12 ) = 56,

and a∗ (0) = 126,

which implies that the possible inner products in ℵ∗ are 0, ± 21 , and ±1. Thus, deﬁning 240 √ Λ∗ = 2zi u∗i : zi ∈ Z , i=1 ∗

it is easy to verify that Λ is an even integral lattice, and √ M (Λ∗ ) = 2 ℵ∗ . Then it follows from Lemma 9.1 that Λ∗ is a direct sum of the lattices An (n ≥ 1), Dn (n ≥ 4), and En (n = 6, 7, 8). In fact, the only lattice of this ∗ type with at least √ ∗240 minimal vectors is E8 . Thus, Λ is isometric to E8 , and therefore 2ℵ is isometric to M (E8 ). Theorem 9.3 is proved. 2 Theorem 9.4 (Bannai and Sloane [1]). There is a unique way (up to isometry) of arranging 196560 nonoverlapping unit spheres in E24 so that they all touch another unit sphere.

9.4. Two Theorems of Bannai and Sloane

145

Let ℵ = {u1 , u2 , . . . , u196560 } be a {24, 196560, π/3} code. We proceed to show that 2ℵ is isometric to M (Λ24 ). In other words, writing Λ =

196560

2zi ui : zi ∈ Z ,

i=1

we try to prove that 2ℵ = M (Λ ) and Λ is isometric to Λ24 . This requires some further preparation. A spherical code ℵ of dimension n is called a spherical l-design if hk (σx) = hk (x) (9.11) x∈ℵ

x∈ℵ

holds for every homogeneous polynomial hk of degree k ≤ l and for every isometry σ of E n . It has the following deﬁning property. Lemma 9.2 (Delsarte, Goethals, and Seidel [1]). Let ℵ be a spherical code of dimension n, cardinality m, and inner product distribution {a(t) : −1 ≤ t ≤ 1}. Then ℵ is a spherical l-design if and only if a(t)Pkα,α (t) = 0 −1≤t≤1

holds for α = (n − 3)/2 and k = 1, 2, . . . , l. Proof. It follows from (9.11) that ℵ is a spherical l-design if % 1 1 hk (σx) = hk (x)dω(x) m nωn bd(Sn )

(9.12)

x∈ℵ

holds for every homogeneous polynomial hk (x) of degree k ≤ l and for every isometry σ of E n . It is well-known that % hk (x)dω(x) = 0 bd(Sn )

holds for every spherical harmonic polynomial hk (x) of degree k ≥ 1. Thus, by (9.12), (8.5), and Assertion 8.7, ℵ is a spherical l-design if and only if hk (x) = 0 (9.13) x∈ℵ

holds for every spherical harmonic polynomial of degree k, 1 ≤ k ≤ l. Let hk,1 , hk,2 , . . . , hk,d(k) be a basis for Hk . By Assertions 8.9 and 8.10, we have d(k) 1 Pkα,α (x, y) = hk,i (x)hk,i (y) ck i=1

146

9. The Kissing Numbers of S8 and S24

for some positive number ck . Hence

a(t)Pkα,α (t) =

−1≤t≤1

1 α,α Pk (x, y) m x, y∈ℵ

=

2 d(k) 1 hk,i (x) . mck i=1 x∈ℵ

Thus, by (9.13), ℵ is a spherical l-design if and only if a(t)Pkα,α (t) = 0 −1≤t≤1

holds for 1 ≤ k ≤ l. Lemma 9.2 is proved. Corollary 9.1.

1 2 M (Λ24 )

2

is a spherical 11-design.

Lemma 9.3 (Bannai and Sloane [1]). The code ℵ is a spherical 11design. Further, letting {a (t) : −1 ≤ t ≤ 1} be its inner product distribution,  1 if t = ±1,     4600 if t = ± 12 ,  a (t) = 47104 if t = ± 14 ,    93150 if t = 0,   0 otherwise. This lemma can be proved by a similar argument to that of the ﬁrst part of the proof of Theorem 9.3. Lemma 9.4 (Bannai and Sloane [1]). Λ is an even integral lattice such that min {v, v : v ∈ Λ \ {o}} = 4. Proof. By Lemma 9.3 it is easy to see that Λ is an even integral √ lattice. Suppose that v, v = 2 holds for some point v ∈ Λ , say v = 2e1 . Then u, v ∈ {0, ±1, ±2},

u ∈ 2ℵ ,

since 2u, v √ = u + v, u + v − u, u − v, v is an even integer and |u, v| ≤ 2 2. In other words, ui , v ∈ 0, ± 21 , ±1 , ui ∈ ℵ . Now we proceed to deduce a contradiction. Suppose ui , v = 0 for m1 choices of ui , |ui , v| = and |ui , v| = 1 for m3 choices. Clearly, m1 + m2 + m3 = 196560.

1 2

for m2 choices,

9.4. Two Theorems of Bannai and Sloane

147

Since ℵ is a spherical 11-design, then by (9.12), % 1 1 hk (ui ) = hk (x)dω(x) 196560 24 ω24 bd(S24 )

(9.14)

ui ∈ℵ

holds for any homogeneous polynomial hk (x) of degree k ≤ 11. Let us choose hk (x) = (x1 )k , for k = 2 and 4, such that hk (ui ) =

ui , v √ 2

k .

By (9.12) and Corollary 9.1, the right-hand side of (9.14) can be evaluated by % 1 1 hk (x)dω(x) = hk (u) 24 ω24 bd(S24 ) 196560 1 u∈ 2 M (Λ24 ) ) 8190 if k = 2, 196560 = 945 if k = 4. 196560 Then, taking k = 2 and 4, respectively, it follows from (9.14) that ) 1 1 8 m2 + 2 m3 = 8190, 1 64 m2

+ 14 m3 = 945,

which implies m2 = 67200 and m3 = −420, an impossibility. Thus, Lemma 9.4 is proved. 2 Corollary 9.2. 2ℵ = M (Λ ). Lemma 9.5 (Bannai and Sloane [1]). Let ℵ be a {24, 196560, π/3} code. For any pair of vectors u and v in ℵ with u , v = 0 there are 44 vectors w ∈ ℵ such that u , w = v , w = 12 .

(9.15)

By Lemma 9.3, the code ℵ is an 11-design. Deﬁne k

hk (x) = f (x)k = (u , x + v , x) . Clearly, hk (x) is a homogeneous polynomial. Also, by Lemma 9.3, f (w ) takes only the eight diﬀerent nonzero values ±1, ± 34 , ± 12 , and ± 41 for w ∈ ℵ , and the number of points w ∈ ℵ \ {u , v } such that f (w ) = 1 is exactly the number of points satisfying (9.15). Considering the cases

148

9. The Kissing Numbers of S8 and S24

k = 1, 2, . . . , 8, this lemma can be proved by a similar argument to that of the proof of Lemma 9.4. Geometrically speaking, Lemma 9.5 means that for any pair of vectors u and v in ℵ with u ov = π/2 there are 44 vectors w ∈ ℵ such that u ow = v ow = π/3. For n ≥ 3 we deﬁne Gn =

n

zi wi : zi ∈ Z ,

i=1

√

√ where w √1 = 2(e1 +e2 ) and wi = 2(ei−1 −ei) for i = 2, 3, . . . , n. In fact, Gn = 2Dn . It is easy to check that w1 ow2 = π/2, wi owi+1 = π/3 (i = 2, 3, . . . , n − 1), G3 ⊂ G4 ⊂ · · · , and there are 2n(n − 1) minimal √ 2 vectors of [ 2 |0n−2 ] type in Gn . Lemma 9.6 (Bannai and Sloane [1]). For n = 3, 4, . . ., 24, the lattice Λ contains a suitable sublattice isometric to Gn . Proof. It follows from Lemma 9.5 that Λ contains a suitable sublattice isometric to G3 . Now we proceed by induction on n. Suppose the assertion holds for n = k, 3 ≤ k < 24. By choosing a suitable orthonormal basis e1 , e2 , . . . , e24 , M (Λ ) contains vectors w1 , w2 , . . . , wk that span Gk . By Lemma 9.5 there are 44 vectors w ∈ M (Λ ) with w1 ow = w2 ow = π/3. On the other hand, it can be veriﬁed that Gk has 2k − 4 < 44 minimal vectors w such that w1 ow = w2 ow = π/3. Thus, at least one of the 44 vectors w, say wk+1 , is not a minimal vector of Gk . Now we proceed to show that wk+1 ∈ E k , where E k indicates the space spanned by w1 , w2 , . . . , wk . Suppose that, on the contrary, wk+1 = w1 e1 + w2 e2 + · · · + wk ek . By w1 owk+1 = w2 owk+1 = π/3 it follows that w1 = For 3 ≤ i ≤ k, √ 2(e1 ± ei ) ∈ M (Λ ) ∩ Gk ⊆ 2ℵ and therefore

√

2 and w2 = 0.

√ wk+1 , 2(e1 ± ei ) ∈ {0, ±1, ±2},

which implies w3 = w4 = · · · = wk = 0. Then, we obtain wk+1 , wk+1 = 2, which contradicts wk+1 ∈ 2ℵ . Thus, we have wk+1 ∈ E k . Choose ek+1 such that {e1 , e2 , . . . , ek+1 } is an orthonormal basis for the (k + 1)-dimensional space E k+1 that contains E k and wk+1 , and suppose that wk+1 = w1 e1 + w2 e2 + · · · + wk+1 ek+1 .

9.4. Two Theorems of Bannai and Sloane The previous √ argument shows that w1 = wk+1 = ± 2. Hence, Gk+1 =

k+1

√

149 2, w2 = w3 = · · · = wk = 0, and

zi wi : zi ∈ Z

⊆ Λ .

i=1

2

Lemma 9.6 is proved.

Proof of Theorem 9.4. By Lemma 9.6 we may choose an orthonormal basis e1 , e2 , . . . , e24 in E 24 such that√G24 ⊂ Λ , and therefore M (Λ ) 2 22 contains √ all the points v of [4 |0 ]/ 8 type. Let u = (u1 , u2 , . . . , u24 )/ 8 be any point of M (Λ ). Then 24

(ui )2 = 32.

(9.16)

i=1

On the other hand, it follows from Lemma 9.3 that u , v = {0, ±1, ±2, ±4}. In other words, 1 2 (ui

± uj ) ∈ {0, ±1, ±2 ± 4}

(9.17)

for all indices i = j. By (9.17) and (9.16) we have ui = 12 (ui + uj ) + 12 (ui − uj ) ∈ {0, ±1, ±2, ±3, ±4, ±5}. Also, it follows from (9.17) that u1 ≡ u2 ≡ · · · ≡ u24

(mod 2)

and, if |ui | = 4 for one index i, u1 ≡ u2 ≡ · · · ≡ u24

(mod 4).

(9.18)

Thus, as a conclusion, the only possibilities for the components of u ∈ M (Λ ) are √ √ √ [28 |016 ]/ 8, [42 |022 ]/ 8, and [31 |123 ]/ 8. Denote by M1 , M2 , and M3 respectively the subsets of M (Λ ) consisting of points of the above types. Now we study the structure of these subsets. Case 1. Let F (x) = (f (x1 ), f (x2 ), . . . , f (x24 )) be a map deﬁned by 0 if x = 0, f (x) = 1 otherwise,

150

9. The Kissing Numbers of S8 and S24

and deﬁne ℵ1 = {F (u ) : u ∈ M1 } . Clearly, the weight of every codeword of ℵ1 is eight. If F (u ) = F (v ) and f (ui ) = f (vi ) = 1 for at least ﬁve indices for two points u and v of M1 , then by (9.18) one can ﬁnd a point w ∈ G24 such that u + v + w < 2. This is impossible. Thus, (

24 8 card{ℵ1 } ≤ = 759. 5 5 √ Also, without loss of generality, √ if u = (u1 , u2 , . . . , u24 )/ 8 ∈ M1 and u1 = 0, then (−ui , u2 , . . . , u24 )/ 8 ∈ M1 . This means that at most 27 diﬀerent points of M1 correspond to one codeword of ℵ1 . Hence, card{M1 } ≤ 759 × 27 = 97152. Case 2. Clearly,

card{M2 } =

24 2 2 = 1104. 2

(9.19)

(9.20)

Case 3. Let G(x) = (g(x1 ), g(x2 ), . . . , g(x24 )) be a map deﬁned by  √ 8,  1 if x = 1/ √ g(x) = 1 if x = −3/ 8,  0 otherwise, and deﬁne ℵ3 = {G(u ) : u ∈ M3 } . Then, routine arguments yield u, vH ≥ 8 for any pair of distinct codewords u and v of ℵ3 . We now proceed to estimate card{ℵ3 } and card{M3 }. Denote the [24, 24] binary code by ℵ, and deﬁne   1 if u, vH < 4, 1 λ(u, v) = if u, vH = 4,  6 0 otherwise. Without loss of generality, if u = (0, 0, . . . , 0), u, vH = u, wH = 4, and v, wH = 8, then vi wi = 0 for all indices i. Thus, it is easy to see that u∈ℵ3

λ(u, v) ≤ 1

9.4. Two Theorems of Bannai and Sloane

151

for every codeword v ∈ ℵ, and therefore λ(u, v) = λ(u, v) ≤ 224 . u∈ℵ3 v∈ℵ

v∈ℵ u∈ℵ3

On the other hand,

λ(u, v) = 1 +

v∈ℵ

24 1

+

24 2

+

24 3

+

1 24 = 212 . 6 4

Then, we have card{ℵ3 } ≤ 224 /212 = 212 . Also, at most 24 points of M3 correspond to one codeword of ℵ3 . Thus, card{M3 } ≤ 24 × 212 = 98304.

(9.21)

From our consideration of these cases, and since 97152 + 1104 + 98304 = 196560, we conclude that equality must hold simultaneously in (9.19), (9.20), and 24 (9.21). Then, by Case 3 and choosing √ an appropriate basis for E ,j Mk3 con-∗ 1 23 ∗ tains all the points of [−3 |1 ] / 8 type, where a point x is of [a |b | · · ·] type if xi = a for j coordinates, xi = b for √ k coordinates, etc. In this case, 1 7 16 ∗ of [−2 |2 |0 ] / 8 type. Otherwise, one could M (Λ ) contains no point √ ﬁnd a point of [124 ]/ 8 type in Λ , which is impossible. Thus, denoting √ by M ∗ the set of 759 points of [28 |016 ]∗ / 8 type corresponding to ℵ1 and using (9.19), we have M ∗ ⊂ M (Λ ). Now √ we proceed to show that M (Λ ) can be generated by u24 = (−3, 8, M ∗ , and G24 . √1, . . . , 1)/ √ Let u = (u1 , u2 , . . . , u24 )/ 8 ∈ Λ be a point of [2k |024−k ]/ 8 type, k ≥ 8. It follows from Case 1 that there is a point √ )/ 8 ∈ M ∗ v = (v1 , v2 , . . . , v24 such that |ui | = |vi | = 2 for l indices with 5 ≤ l ≤ 8. Then one √ can ﬁnd a point w ∈ G24 such that k+8−2l 16+2l−k u + v + w is of [2 |0 ]/ 8 type. Clearly, k + 8 − 2l < k. By reduction, u can be generated by M ∗ and G24 . From this it is clear that M (Λ ) is generated by u24 , M ∗ , and G24 . Comparing this with the basis of Λ24 deﬁned in Section 1, it follows that M (Λ ) is isometric to M (Λ24 ). Theorem 9.4 is proved. 2

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10. Multiple Sphere Packings

10.1. Introduction A family Sn +X of unit spheres is said to form a k-fold packing in En if each point of the space belongs to the interiors of at most k spheres of the family. In particular, when X is a lattice, the corresponding family is called a kfold lattice packing. Let m(Sn , k, l) be the maximal number of unit spheres forming a k-fold packing and contained in lIn . Then analogously to the densities of classical sphere packings we deﬁne δk (Sn ) = lim sup l→+∞

m(Sn , k, l)v(Sn ) v(lIn )

and δk∗ (Sn ) = sup Λ

v(Sn ) , det(Λ)

where the supremum is over all lattices Λ such that Sn +Λ is a k-fold lattice packing in E n . Usually, δk (Sn ) and δk∗ (Sn ) are called the density of the densest k-fold sphere packings and the density of the densest k-fold lattice sphere packings, respectively. Clearly, one can deﬁne δk (K) and δk∗ (K) in a similar way for a general convex body K. Also, for these densities, it is easy to see that both k δ(K) ≤ δk (K) ≤ k (10.1) and k δ∗ (K) ≤ δk∗ (K) ≤ δk (K)

(10.2)

154

10. Multiple Sphere Packings

hold for every convex body K and for every index k. In this chapter we will discuss some basic results about the densities δk (Sn ) and δk∗ (Sn ). Remark 10.1. Obviously, k-fold packings are not always packings. Nevertheless, this generalization is very natural and has attracted the attention of many geometers.

10.2. A Basic Theorem of Asymptotic Type When n is ﬁxed and k → ∞, we have the following basic theorem describing the asymptotic behavior of δk (Sn ) and δk∗ (Sn ). Theorem 10.1. δk (Sn ) δ ∗ (Sn ) = lim k = 1. k→∞ k→∞ k k lim

Although this is rather counterintuitive when compared with known results about δ(Sn ) and δ∗ (Sn ), this theorem itself is easy to prove. Nevertheless, for the convenience of further generalization, we will deduce it from some stronger and more general results. Lemma 10.1. Let K be a convex body, Λ a lattice, and let D be a polytope such that D + Λ is a tiling of En . If there exist k points p1 , p2 , . . . , pk in Λ such that k (D + pi ), (10.3) K⊂ i=1

then δk∗ (K) ≥

v(K) . v(D)

Proof. If a point x ∈ En belongs to the interiors of m translates of K in K + Λ, say K + q1 , K + q2 , . . . , K + qm , then x − qj ∈ int(K) holds for j = 1, 2, . . . , m. By (10.3) and (10.4) it follows that x − q1 ∈ D + pi for some index i, 1 ≤ i ≤ k. Hence, writing Dj = D + pi + q1 − qj ,

(10.4)

10.2. A Basic Theorem of Asymptotic Type

155

it follows that x − qj ∈ Dj for j = 1, 2, . . . , m. Therefore, by (10.4) we have Dj ∩ int(K) = ∅ for j = 1, 2, . . . , m. On the other hand, it is clear that Dj is a tile in the tiling D + Λ. Thus, each Dj is some D + pi , and therefore m ≤ k. It follows that K + Λ is a k-fold lattice packing with density v(K)/v(D). Lemma 10.1 is proved. 2 Lemma 10.2.

δk∗ (K) ≥ k − c(K)k(n−1)/n ,

where c(K) is a positive constant depending only on K. Proof. As usual, we denote the Minkowski sum of two convex bodies K1 and K2 by K1 + K2 . In other words, K1 + K2 = {x + y : x ∈ K1 , y ∈ K2 } . We will require Steiner’s formula, that v(K1 + tK2 ) = v(K1 ) +

n

n Wi (K1 , K2 )ti i i=1

(10.5)

holds for all positive numbers t, where Wi (K1 , K2 ) (the ith mixed volume of K1 and K2 ) are positive constants depending only on K1 , K2 , and i. In particular, Wn (K1 , K2 ) = v(K2 ). Taking Λ = tZn and D = tIn , it is obvious that D + Λ is a tiling and the tiles that intersect K are contained in K + 2D. Thus, by (10.5), K is covered by 6 7 6

7 n v(K + 2D) v(K) i n i−n = + 2 W (K, I )t i n v(D) tn i i=1 tiles of the tiling. Write

n v(K) i n f (t) = n + 2 Wi (K, In )ti−n . t i i=1

Clearly, the function f (t) is continuous on (0, ∞), lim f (t) = ∞,

t→0

156

10. Multiple Sphere Packings

and lim f (t) = 2n .

t→∞

Thus, for every large number k there exists a positive number tk such that

n v(K) i n k= n + 2 Wi (K, In )ti−n k . tk i i=1

(10.6)

By Lemma 10.1, K + tk Zn forms a k-fold lattice packing with density

n v(K) i n = k − 2 Wi (K, In )ti−n k . tnk i i=1

(10.7)

By (10.6), we have k≥ and therefore

tk ≥

v(K) tnk

v(K) k

1/n .

(10.8)

Then, it follows from Lemma 10.1, (10.7), and (10.8) that δk∗ (K) ≥

v(K) v(K) = n ≥ k − c(K)k(n−1)/n , v(D) tk

where c(K) is a suitable positive constant depending only upon K. Lemma 10.2 is proved. 2 Proof of Theorem 10.1. Theorem 10.1 is a direct consequence of (10.1), (10.2), and Lemma 10.2. 2 Remark 10.2. Theorem 10.1 can be found in Bolle [1] and in Groemer [4]. In fact, this theorem is true not only for spheres but also for every convex body, and even for any bounded subset of En with positive Jordan measure. Remark 10.3. In the two-dimensional case, using a more explicit method, Bolle [3] proved k − c1 k 2/5 ≤ δk∗ (S2 ) ≤ k − c2 k 1/4 , where c1 and c2 are suitable positive constants. Remark 10.4. For general n and k, by modifying Blichfeldt’s method, Few [3] and [6] proved

n/2 1 k δk (Sn ) ≤ (n + 1)k − 1 1 + n k+1

10.3. A Theorem of Few and Kanagasahapathy and, in particular,

157

n/2 4(n + 2) 2 . δ2 (Sn ) ≤ 3 3

10.3. A Theorem of Few and Kanagasahapathy Theorem 10.2 (Few and Kanagasahapathy [1]). 8π δ2∗ (S3 ) = √ . 9 3 In addition, the optimal lattice is unique up to isometry. √ √ Taking Λ∗ to be the lattice with basis a1 = ( 3, 0, 0), a2 = ( 3/2, 3/2, ∗ 0), and a3 = (0, 0, 1), routine √ computation shows that S3 + Λ is a double packing with density 8π/9 3. Thus, 8π δ2∗ (S3 ) ≥ √ . 9 3

(10.9)

Let Λ be any lattice in E3 such that S3 + Λ forms a double packing. We proceed to show that √ 3 3 det(Λ) ≥ , (10.10) 2 where equality holds only if Λ is isometric to Λ∗ . We will require the following two technical lemmas concerning Λ. Lemma 10.3. Let opq, where p, q ∈ Λ, be an acute-angled triangle in which o, p ≤ 2 is the shortest side. Assume that p = (p, 0, 0) and q = (α, q, 0), where α, p, and q are all positive. Then, we have p ≥ 1, o, q2 + p, q2 ≥ 6, q ≥ 4 − p2 , q≥ and

p2 4 − p2 , 2

√ 3p q≥ , 2

(10.11) (10.12)

2 ≤ p2 ≤ 3,

p2 ≥ 3,

where equality in (10.12) holds only if α = 0 or α = p.

(10.13)

(10.14)

158

10. Multiple Sphere Packings

Proof. First, we must have p ≥ 1, since otherwise o would be covered three times by the interiors of S3 − p, S3 , and S3 + p. Take x = (p/2, 4 − p2 /2, 0). Then q ∈ int(S3 ) + x, since otherwise some point near x would be covered three times. Hence, (10.12) follows from the fact that opq is an acute-angled triangle.

(0,

4 − p2 , 0)

(p,

4 − p2 , 0)

x

o

p Figure 10.1

2 2 If p2 ≥ 3, then (10.11) is trivial,and if p2 ≤ 2, then o, q + p, q 2 2 attains its minimum when q = (0, 4 − p , 0) or q = (p, 4 − p , 0) (see Figure 10.1). In either case

o, q2 + p, q2 = 4 + (4 − p2 ) ≥ 6. Suppose now that 2 ≤ p2 ≤ 3. In this case both o, q2 + p, q2 and q attain their minima when o, q = p and q, x = 1 (see Figure 10.1). Writing θ = poq, we have θ 4 − p2 θ p sin = , cos = , 2 2 2 2 and q ≥ p sin θ = 2p sin

θ θ p2 cos = 4 − p2 , 2 2 2

which proves (10.13). Also, p, q = 2p sin

θ = p 4 − p2 2

and o, q2 + p, q2 − 6 = p2 + p2 (4 − p2 ) − 6 = (2 − p2 )(p2 − 3) ≥ 0, which completes the proof of (10.11).

10.3. A Theorem of Few and Kanagasahapathy

159

The result (10.14) is trivial, since o, q ≥ p and p, q ≥ p. Lemma 10.3 is proved. 2 Lemma 10.4. Let y be any point in the parallelogram with vertices o, p, 1 1 2 (p + q), and 2 (q − p). Then either y, 12 p ≤ 12 o, q or

y, 12 q ≤ 12 o, p.

Proof. Let v be the perpendicular foot from 12 (p+q) to the line determined by o and q − p. Then, since o, q ≥ o, p, v is between o and 12 (q − p) (see Figure 10.2). Since (q − p)v(p + q) = π/2 and 12 q is the midpoint of 12 (q − p) and 12 (p + q), we have v, 12 q = 12 o, p. Similarly, since 12 p is the midpoint of 12 (p + q) and 12 (p − q), v, 12 p = 12 o, q. q

1 2 (q

1 2 (p

− p)

+ q)

v

o

p

Figure 10.2 Hence, if y belongs to the triangle with vertices 12 (p + q), 12 (q − p), and v, we have y, 12 q ≤ 12 o, p, and if y belongs to the quadrilateral with vertices v, o, p, and 12 (p + q), we have y, 12 p ≤ 12 o, q. Lemma 10.4 is proved.

2

Proof of Theorem 10.2. First we try to prove (10.10). Let p be one of the nearest points of Λ to o. Then o, p ≥ 1, since otherwise o would be covered three times by the interiors of S3 − p, S3 , and S3 + p. On the

160

10. Multiple Sphere Packings

other hand, we may suppose that o, p < 2; otherwise, S3 + Λ is a normal packing. Let L denote the line determined by o and p. Let q1 ∈ Λ \ L be a point such that the distance from q1 to L is minimal, and let q2 ∈ Λ \ L be a point such that o, q2 = r is minimal. We may take p = (p, 0, 0) with 1 ≤ p < 2, and qi = (αi , qi , 0) with qi ≥ 0 and 0 ≤ αi < p/2. Since op is the shortest side of the triangle opqi , and 0 ≤ αi < p/2, it follows that the triangles opq1 and opq2 are acute-angled. (We are not assuming that o, p, q1 , and q2 are coplanar. We make two choices of axes of coordinates, one for each basis.) Let ui (i = 1, 2) be chosen such that p, qi , and ui generate Λ. We may take ui = (ai , bi , ci ), where the corresponding points u∗i = (ai , bi , 0) belong to the parallelogram with vertices o, p, 12 (p + qi ), and 12 (qi − p) (see Figure 10.3). qi

1 2 (qi

ui

− p)

1 2 (p

+ qi )

u∗i o

p Figure 10.3

Based on the choices of q1 and u1 it follows that b21 + c21 ≥ q12 ,

0 ≤ b1 ≤

and therefore c21 ≥ q12 − b21 ≥ Thus, we have

3q12 . 4 √

det(Λ) = pq1 c1 ≥

q1 , 2

3pq12 . 2

(10.15)

We now consider ﬁve cases. Case 1. p ≤ 1.3. By (10.15), (10.12), and routine computation it follows that √ √ 3p(4 − p2 ) 3 3 det(Λ) ≥ ≥ , 2 2

10.3. A Theorem of Few and Kanagasahapathy

161

where equalities hold if and only if the considered lattice Λ is isometric to Λ∗ . Case 2. 2.1 ≤ p2 ≤ 3. By (10.15), (10.13), and routine computation we have √ 5 √ 3 3 3p (4 − p2 ) det(Λ) ≥ > . 8 2 Case 3. 3 ≤ p2 ≤ 4. By (10.15) and (10.14), √ √ 3 3 3 27 3 3 det(Λ) ≥ p ≥ > . 8 8 2 √ Case 4. 1.3 ≤ p ≤ 2.1 and r2 ≥ 3. Let w2 be the lattice point in the plane {x : x3 = c2 } that is nearest to o. Then, o, w2 2 ≤ c22 +

p2 p2 q2 r2 + 2 ≤ c22 + + . 4 4 4 4

Hence, based on the choice of q2 , c22 +

p2 r2 + ≥ r2 , 4 4

3r2 − p2 9 − p2 ≥ . 4 4 Therefore, by (10.12) and routine analysis we have c22 ≥

√ p 3 3 2 2 (4 − p )(9 − p ) > . det(Λ) = pq2 c2 ≥ 2 2 √ Case 5. 1.3 ≤ p ≤ 2.1 and r2 ≤ 3. The triangle oq2 u2 is acute-angled and oq2 is the shortest side. Hence, by elementary geometry and (10.11), o, u2 2 + u2 , q2 2 = 2c22 + o, u∗2 2 + q2 , u∗2 2 = 2c22 + 2u∗2 , 12 q2 2 + 12 r2 ≥ 6.

(10.16)

We now need to prove that c22 ≥

9 − p2 . 4

(10.17)

If u∗2 , 12 q2 ≤ p/2, then (10.17) follows from (10.16). If u∗2 , 12 q2 > p/2, it follows from Lemma 10.4 that u∗2 , 12 p ≤ 12 r.

(10.18)

162

10. Multiple Sphere Packings

If the angle u2 op is obtuse, then the angle sop, where s = u2 + p, is acute and s∗ , 12 p ≤ u∗2 , 12 p ≤ 12 r,

(10.19)

where s∗ = u∗2 + p. Hence applying (10.11) to the triangle opt, where t = u2 or s, o, t2 + t, p2 = 2c22 + o, t∗ ∗ + p, t∗ 2 = 2c22 + 2t∗ , 12 p2 + 12 p2 ≥ 6,

(10.20)

where t∗ = u∗2 or s∗ . In this case, (10.17) follows from (10.18), (10.19), and (10.20). Then, by (10.12) and (10.17), √ p 3 3 2 2 det(Λ) = pq2 c2 ≥ (4 − p )(9 − p ) > . 2 2 Hence, (10.10) is proved. Clearly, (10.9) and (10.10) together yield Theorem 10.2. 2 Remark 10.5. Theorem 10.2 is the only known result about the exact values of δk (Sn ) and δk∗ (Sn ) when n ≥ 3 and k ≥ 2.

10.4. Remarks on Multiple Circle Packings In E 2 , much more explicit results are available. Since their proofs are elementary and technical, we only quote the results. k

δk∗ (S2 )

Author

2

π √ 3 3π 2

Heppes [1]

2π √ 3 2π √ 7 35π √ 8 6 8π √ 15 3969π √ √ √ 4 (220−2 193)(449+32 193)

Heppes [1]

3 4 5 6 7 8

Heppes [1]

Blundon [2] Blundon [2] Bolle [2] Yakovlev [1]

10.4. Remarks on Multiple Circle Packings

163

Several Vienna dissertations dealt with problems of this kind. Unfortunately, it is not easy to obtain them. Also, Linhart [1] and Temesv´ ari, Horv´ ath, and Yakovlev [1] developed algorithms with which one can determine δk∗ (S2 ) for each k to any prescribed accuracy. Multiple packings of general 2-dimensional convex domains have been studied by many authors. For results of this kind we refer to G. Fejes T´ oth and W. Kuperberg [1].

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11. Holes in Sphere Packings

11.1. Spherical Holes in Sphere Packings In order to study the eﬃciency of a sphere packing, it is both important and interesting to investigate its holes, especially spherical holes. Let Sn +X be a sphere packing in E n , and let r(Sn , X) be the supremum of the radii of all spheres disjoint from any sphere of the packing. The number 1/r(Sn , X) is called the closeness of the packing Sn + X. We deﬁne r(Sn ) = inf {r(Sn , X)} , X

where the inﬁmum is taken over all sets X such that Sn + X is a packing. By routine argument it follows that there exists a discrete set X such that r(Sn ) = r(Sn , X). The corresponding packing Sn + X is called the closest sphere packing. Similarly, one can deﬁne r∗ (Sn ) by assuming that X is a lattice. Clearly, we have r(Sn ) ≤ r∗ (Sn ). These concepts were ﬁrst introduced by L. Fejes T´oth [11], who suggested the problems of determining the number r(Sn ) and the corresponding closest sphere packings. These problems, like Kepler’s conjecture and the Gregory-Newton problem, are both natural and important. However, our knowledge about them is comparatively limited. In this section we will prove the following main result. Theorem 11.1 (B¨ or¨ oczky [2]). r(S3 ) = r∗ (S3 ) =

5/3 − 1.

166

11. Holes in Sphere Packings

Also, up to isometry, r(S3 , X) = r(S3 ) = r∗ (S3 ) if and only if X is a suitable space-centered cubic lattice. To prove this theorem, we need several technical lemmas. Lemma 11.1 (B¨ or¨ oczky [2]). Let T = v1 v2 v3 v4 be a tetrahedron with circumradius r such that vi , vj ≥ 2, i = j, and the dihedral angle at edge v1 v2 is at most π/3. Then r ≥ 5/3, where equality holds if and only if T is congruent to the tetrahedron T = v1 v2 v3 v4 for which √ v1 , v3 = v2 , v4 = 4/ 3 and v1 , v2 = v2 , v3 = v3 , v4 = v4 , v1 = 2. Proof. Let S be the circumsphere of T centered at o. Lemma 11.1 will be proved by showing that any tetrahedron T inscribed in S and satisfying the conditions of the lemma is congruent to T .

v1 w1

v4 w2

v3

w3

w4 v2 Figure 11.1

Assume that v1 v2 is parallel to v1 v2 , and the four points v1 , v1 , v2 , and v2 are contained in a half great circle of S . Denote by G the part of bd(S ) cut out by the dihedral angle of T at the edge v1 v2 , by H the spherical set {x ∈ bd(S ) : x, v1 ≥ 2, x, v2 ≥ 2}, and by Q the intersection of G and H. Clearly, we have {v3 , v4 } ⊂ Q,

(11.1)

11.1. Spherical Holes in Sphere Packings

167

and Q is a spherical domain with four vertices, say w1 , w2 , w3 , and w4 in a circular order. Also, d(Q) = w1 , w3 = w2 , w4 ,

(11.2)

where {w1 , w3 } and {w2 , w4 } are the only solutions (see Figure 11.1). Since (11.1) and (11.2) hold, we may modify the considered tetrahedron as follows. Step 1. Replace v1 and v2 by v1 and v2 , respectively. Then, replace v3 and v4 by the new points w1 and w3 , respectively. Step 2. Move v3 and v4 along the boundary of H until the dihedral angle of T at v1 v2 is π/3. Clearly, d(Q) strictly increases during this process. v1

v4

m

o4

o3 v3

v2 Figure 11.2 Let T be a modiﬁed tetrahedron, m the midpoint of v1 v2 , oi the center of the circumcircle of v1 v2 vi , and write x = m, o3 and y = m, o4. Then, elementary but complicated computation (see Figure 11.2) yields that √ 3+1 1 ≤ xy ≤ 6 2 and 96(1 + xy) v3 , v4 2 = 36 − 2 2 ≤ 4, 2x y + 2xy + 3 where equality holds if and only if T is congruent to T . Hence, Lemma 11.1 is proved. 2 √ Write b = 4/ 3 and let Λ be the space-centered cubic lattice with basis u1 = (b, 0, 0), u2 = (0, b, 0), and u3 = (b/2, b/2, b/2). Then, T is congruent to the tetrahedron with vertices u1 , u1 + u2 , u3 , and u3 + u1 . Also, since the Dirichlet-Voronoi cells of Λ are regular truncated octahedra, the solid

168

11. Holes in Sphere Packings

angle formed by the three planes perpendicular to the edges of T meeting at a vertex is congruent to the solid angle at a vertex of a regular truncated octahedron. Lemma 11.2 (B¨ or¨ oczky [2]). There is no convex polyhedron bounded by one quadrangle and ten pentagons such that in each vertex of the polyhedron exactly three facets meet. Proof. Suppose there is such a polyhedron, say P . We proceed to deduce a contradiction. Let v1 , v2 , v3 , and v4 be the vertices of the quadrangular facet of P , in a circular order (see Figure 11.3). Let wi be the endpoint of the third edge of P emanating from vi , i = 1, . . . , 4. The points wi are diﬀerent from one another, and diﬀerent from the points vi , i = 1, . . . , 4. For convenience, we write v5 = v1 and w5 = w1 . Then, for 1 ≤ i ≤ 4, the points wi , vi , vi+1 , and wi+1 are four consecutive vertices of a pentagonal facet of P . Let xi be the ﬁfth vertex of this facet. Again, the points vi , wi , xi , i = 1, . . . , 4, are all distinct. In the part of P considered so far the points vi and wi have valency 3, while xi has valency 2. Thus, there is a third vertex, say yi , that is connected by an edge to xi . Again for convenience, we write x5 = x1 and y5 = y1 . It is easy to see that yi = xi+1 and yi = yi+1 , since otherwise we would obtain a triangular or quadrangular facet. If we have yi = xi+2 for i = 1 or 2, then yi+1 has valency 1; if we have yi = yi+2 for i = 1 or 2, then yi = yi+1 = yi+2 . In the second case, xi wi+1 xi+1 yi and yi xi+1 wi+2 xi+2 are quadrangular facets of P . Hence, the points yi are diﬀerent from one another and from the points vi , wi , and xi , i = 1, . . . , 4. y1

y2 w2

x1

x2

v2 w1

v3

v1

w3

v4 x4

w4

x3 y3

y4 Figure 11.3

Now, yi , xi , wi+1 , xi+1 , and yi+1 are ﬁve consecutive vertices of a facet of P ; thus the points yi and yi+1 are connected by an edge. This means that all the vertices considered so far are trivalent, and our polyhedron has only ten facets, two quadrangles, and eight pentagons. By this contradiction Lemma 11.2 is proved. 2

11.1. Spherical Holes in Sphere Packings

169

It is well-known (see Coxeter [5]) that there are eight diﬀerent types of regular polytopes in E 4 , one of which, say P , is bounded by 120 regular dodecahedra such that at each vertex four of them meet. A cell complex {F, ≺} is a family F of cells (polytopes and their faces) with an incidence relation ≺ (for example, the relation ⊂). Two cell complexes {F1 , ≺1 } and {F2 , ≺2 } are isomorphic if there is a one-to-one mapping f : F1 → F2 such that f (X) ≺2 f (Y ) if and only if X ≺1 Y for X, Y ∈ F1 . Lemma 11.3 (B¨ or¨ oczky [2]). There is no tetravalent facet-to-facet tiling in E 3 such that each tile has exactly 12 pentagonal facets. Proof. Let ℘ be the family of all faces of P . Then, {℘, ⊂} is a cell complex. Without loss of generality, we label the facets of P as F1 , F2 , . . . , F120 such that for k = 1, 2, . . . , 119, the set Uk =

k

Fi

i=1

is homeomorphic to S3 and the set Uk = Fk ∩

k−1

Fi

i=1

is homeomorphic to S2 . For convenience, we write ℘k = {X ∈ ℘ : X ⊂ Uk } , ℘k = X ∈ ℘ : X ⊂ Uk+1 , and ℘120 = ℘. Suppose, on the contrary, there is a tetravalent facet-to-facet tiling in E3 such that each tile has exactly 12 pentagonal facets. Let P1 be an arbitrary tile of the tiling, let 1 be the family consisting of P1 and all its faces, and let ≺1 be the relation ⊂. Clearly, the two cell complexes {℘1 , ⊂} and {1 , ≺1 } are isomorphic. Let f1 be an isomorphism between them. Now, for k = 1, 2, . . . , 120, we proceed to deﬁne cell complexes {k , ≺k } and isomorphisms fk between {℘k , ⊂} and {k , ≺k } by induction. In general, the cell complex {k , ≺k } has the following properties: 1. k is a family consisting of some tiles of the tiling and all their faces. 2. If X ≺k Y , then X ⊂ Y . Suppose that we have a cell complex {k , ≺k } with the above properties and an isomorphism fk between {℘k , ⊂} and {k , ≺k }. Let {k , ≺k } be

170

11. Holes in Sphere Packings

the subcomplex of {k , ≺k } corresponding to {℘k , ⊂} at the isomorphism fk , let Qk ∈ k be a facet of some tile, say Pk ∈ k , and let Pk+1 be the tile such that Qk = Pk ∩ Pk+1 . Since Uk+1 is homeomorphic to S2 , all the facets in k are connected by edges of the tiles. On the other hand, since the tiling is tetravalent and is facet-to-facet, at every edge three tiles join properly together. Thus, Pk+1 is independent of the choice of Qk . Let k+1 be the family consisting of k , Pk+1 , and all the faces of Pk+1 , and deﬁne the relation ≺k+1 by X ≺k Y if X, Y ∈ k , X ≺k+1 Y = X ⊂Y if Y ∈ k+1 \ k . Then, it is easy to show that the cell complex {k+1 , ≺k+1 } satisﬁes the required properties and is isomorphic to {℘k+1 , ⊂}. As a result of this process, we obtain a cell complex {120 , ≺120 } that satisﬁes properties 1 and 2 and is isomorphic to {℘, ⊂}. Obviously, this contradicts the fact that P1 , P2 , . . . , P120 are tiles of the tiling. Thus, Lemma 11.3 is proved. 2 Now we introduce a fundamental result concerning cap packings and cap coverings in bd(S3 ) that will be useful in the proof of Theorem 11.1. Lemma 11.4 (L. Fejes T´ oth [9]). Let γ(m) be the maximum spherical radius such that m caps of such radius can be packed into bd(S3 ). Let γ (m) be the minimum spherical radius such that m caps of such radius can cover bd(S3 ), and write m θm = π. 6(m − 2) Then, γ(m) ≤ arccos and γ (m) ≥ arccos

1

2 cosec θm

"

1 3 cot θm

,

where equalities hold only if m = 3, 4, 6, or 12. Proof. Let Ω1 , Ω2 , . . . , Ωm be m caps, of spherical radius γ, centered at o1 , o2 , . . . , om , respectively, and let D1 , D2 , . . . , Dm be the corresponding spherical Dirichlet-Voronoi cells. It is easy to see that Ωi ⊂ Di if the caps form a packing in bd(S3 ), Di ⊂ Ωi if they form a covering in bd(S3 ), and that the Dirichlet-Voronoi cells form a tiling in bd(S3 ). For convenience, we assume that the Ωi are inscribed in the Di in the packing case and that the Di are inscribed in the Ωi in the covering case. Let k be the number of edges of the tiling. By Euler’s formula and the fact that at least three edges meet at every vertex, it can be deduced that k ≤ 3(m − 2).

(11.3)

11.1. Spherical Holes in Sphere Packings

171

Joining each oi to the vertices of Di and its perpendicular feet on the edges of Di by arcs, bd(S3 ) is divided into 4k spherical triangles. Consider one of them, say T , and assume that its spherical angle at the corresponding oi is α. It is well-known that s(T ) = α − arcsin(cos γ · sin α)

(11.4)

in the packing case, and s(T ) = α − arctan(cos γ · tan α)

(11.5)

in the covering case. Since (11.4) is a convex function of α and (11.5) is a concave function of α when 0 < α < π/2, by Jensen’s inequalities we have mπ mπ 4π = T ≥ 4k − arcsin cos γ · sin 2k 2k and therefore cos γ ≥

sin (m−2)π 2k sin mπ 2k

in the packing case. Also, mπ mπ 4π = T ≤ 4k − arctan cos γ · tan , 2k 2k and hence cos γ ≤

tan (m−2)π 2k tan mπ 2k

in the covering case. Thus, by (11.3) we have cos γ(m) ≥

sin(π/6) 1 = cosec θm sin θm 2

(11.6)

tan (π/6) 1 = √ cot θm tan θm 3

(11.7)

in the packing case, and cos γ (m) ≤

in the covering case. Checking the cases of equality in every step, it is easy to see that equalities hold in (11.6) and (11.7) only if m = 3, 4, 6, or 12. Thus, Lemma 11.4 follows from (11.6) and (11.7). 2 With these preparations, Theorem 11.1 can be proved as follows. Proof of Theorem 11.1. For convenience, we write τ = 5/3. In E 3 , let X = {x1 , x2 , . . . } be a discrete set such that S3 + X is a packing and τ S3 + X is a covering, and let Di be the Dirichlet-Voronoi cell associated with xi . Obviously, we have S3 + xi ⊂ Di ⊂ τ S3 + xi ,

i = 1, 2, . . . .

(11.8)

172

11. Holes in Sphere Packings

First, we proceed to show that the tiling formed by the Dirichlet-Voronoi cells is tetravalent. In other words, at every vertex four Dirichlet-Voronoi cells meet. Suppose, on the contrary, a vertex v belongs to ﬁve cells, say D1 , D2 , . . . , D5 . Then xi ∈ bd(dS3 ) + v,

i = 1, 2, . . . , 5,

where d is the distance between v and x1 . It follows from routine combinatorial argument that two of these ﬁve points, say x1 and x2 , satisfy

x1 vx2 ≤ π/2.

Thus, since x1 , x2 ≥ 2, we obtain d = v, x1 ≥ cosec (π/4) =

√

2>

5/3 = τ,

which contradicts (11.8). As a simple consequence, at every vertex of every Dirichlet-Voronoi cell three edges meet. x2

v3 F v2 v1 x1 Figure 11.4 Second, we claim that none of the Dirichlet-Voronoi cells has a facet with more than six vertices. Suppose, on the contrary, that one of the cells, say D1 , has a facet F with more than six vertices v1 , v2 , . . . , vl , and assume that v2 is also a vertex of D2 , D3 , and D4 ,

v1 v2 v3 > 2π/3,

and x1 x2 is perpendicular to F . Then the dihedral angle of the tetrahedron x1 x2 x3 x4 at the edge x1 x2 is (see Figure 11.4) π − v1 v2 v3 < π/3. Thus, Lemma 11.1 implies x1 , v2 > τ,

11.1. Spherical Holes in Sphere Packings

173

which contradicts (11.8). Next, we are going to show that one of the Dirichlet-Voronoi cells, say D1 , has a hexagonal facet. Assume that D1 has k facets with e1 , e2 , . . . , ek edges, respectively. It follows from Euler’s formula and the fact that three edges meet at every vertex that k

ei = 6k − 12.

i=1

Thus, keeping the second claim in mind, the assertion can be easily shown if k > 12. We proceed to deduce the assertion by excluding the following cases. Case 1. k ≤ 10. It follows from Lemma 11.4 that there is a vertex v of D1 such that v, x1 ≥ sec γ (10) > τ, which contradicts (11.8). Case 2. k = 11 and ei ≤ 5 for i = 1, 2, . . . , 11. By Euler’s formula we have, without loss of generality, e1 = 4 and e2 = · · · = e11 = 5. Then, by Lemma 11.2 this is impossible. Case 3. Each of the Dirichlet-Voronoi cells has exactly 12 facets, each of which has at most 5 edges. Then, Euler’s formula implies that every facet of every cell is a pentagon. This case is excluded by Lemma 11.3. Let F1 = v1 v2 . . . v6 be a hexagonal facet of D1 . Since no angle of F1 can be greater than 2π/3, every angle of F1 is 2π/3. Without loss of generality, we may assume that F1 = D1 ∩ D2 and let x2+i , i = 1, 2, . . . , 6, be the points such that vi vi+1 is the radical axis of x1 , x2 , and xi+2 , where v7 = v1 . Then, the six tetrahedra x1 x2 x3 x4 , x1 x2 x4 x5 , . . . , x1 x2 x8 x3 satisfy the condition of Lemma 11.1, and their circumradii are not greater than τ . Thus all of them are congruent to T and, in particular, x1 , x2 = 2 and √ x1 , x3 = x1 , x5 = x1 , x7 = 4/ 3. (11.9) For convenience, we denote the midpoint of x1 xi by mi. Then, by routine computation it follows that m2, vi = 2/3, i = 1, 2, . . . , 6, and therefore, since every angle of F1 is 2π/3, F1 is a regular hexagon. By the observation just above Lemma 11.2, we can draw a regular truncated octahedron G of side length 2/3 and centered at x1 , say x1 = o, such that F1 is a facet of G and the facets of G and D1 adjacent to F1 lie in the same planes. Without loss of generality, we may assume that the three facets of G along the edges v1 v2 , v3 v4 , and v5 v6 are squares.

174

11. Holes in Sphere Packings

We proceed to show that D1 = G. First, we prove that the three facets F2 , F4 , and F6 of D1 that join F1 at edges v1 v2 , v3 v4 , and v5 v6 , respectively, are squares (see Figure 11.5). By (11.9) and Lemma 11.1, every angle of F2 is less than 2π/3. This, together with the fact that the angles of F2 at v1 and v2 are π/2, implies that F2 is a quadrangle, say v1 v2 w2 w1 . Let x9 be the point such that the line w1 w2 is the radical axis of x1 , x1 x3 x9 . Since x1 , x9 ≥ 2, x3 , and x9 . Now, we consider the triangle √ x3 , x9 ≥ 2, and x1 , x3 = b = 4/ 3, we have x1 x9 x3 < π/2. Thus, routine analysis yields that the distance from the center of the circumcircle of x1 x3 x9 to the line x1 x3 attains its minimum 1/6 when x1 , x9 = x3 , x9 = 2. This means that the line w1 w2 does not intersect the open circle C = x ∈ H2 : x, m3 < 1/6 , where H2 indicates the plane determined by F2 and mi indicates the midpoint of x1 xi . On the other hand, the distance between x1 and any point x ∈ H2 that does not belong to the circle C = x ∈ H2 : x, m3 ≤ 1/3 is greater than τ . Thus, both w1 and w2 belong to C . We observe that C and C are, in fact, the incircle and circumcircle of the corresponding square facet F2 of G. Therefore, we have F2 = F2 . x2 x3 F1 v3

m2 v2

v1 m3

F2 x9

m9

x1 = o Figure 11.5

Let F3 be the facet of D1 joining F1 at v2 v3 , and let v2 , v3 , u1 , u2 , u3 , and w2 be its consecutive vertices. Then, v2 , w2 ∈ F2 and v3 , u1 ∈ F4 . By considering the angles at u1 and w2 , it follows that the edges u1 u2 and u3 w2 lie on the lines determined by the corresponding edges of G. If

11.1. Spherical Holes in Sphere Packings

175

u2 = u3 , routine computation yields that x1 , u2 > τ. Hence, F3 is a regular hexagon. Repeating this process, it follows that D1 = G. Then, by considering the neighbors of D1 , we obtain that Di = G + xi and therefore X is a space-centered cubic lattice. Theorem 11.1 follows. 2 Clearly, we have r(Sn ) < 1 in every dimension. We also have the following lower bound for r(Sn ). Theorem 11.2 (Ry˘skov and Horv´ ath [1]). 8 r(Sn ) ≥

n

θ(Sn ) − 1 0.51466. δ(Sn )

Proof. Let Sn + X be a sphere packing in which the maximal radius of the spherical holes is r(Sn ). Then (1 + r(Sn ))Sn + X is a covering in E n . From the deﬁnitions of δ(Sn ) and θ(Sn ) it follows that card{X ∩ lIn }v((1 + r(Sn ))Sn ) v(lIn ) card{X ∩ lIn }v(Sn ) n = lim (1 + r(Sn )) l→∞ v(lIn ) n ≤ (1 + r(Sn )) δ(Sn ).

θ(Sn ) ≤ lim

l→∞

In other words,

8 r(Sn ) ≥

n

θ(Sn ) − 1. δ(Sn )

Also, by Theorem 3.4, Corollary 7.2, and Theorem 8.2 we have 8 θ(Sn ) r(Sn ) ≥ n − 1 0.51466. δ(Sn ) 2

Theorem 11.2 is proved. To end this section, let us list a couple of open problems. Problem 11.1. Is it true that for all n ≥ 2, r(Sn+1 ) ≥ r(Sn )? Problem 11.2. Determine the value of lim r(Sn ). n→∞

176

11. Holes in Sphere Packings

11.2. Spherical Holes in Lattice Sphere Packings Let Sn + Λ be a lattice sphere packing in E n , and write µ(Λ) = r(Sn , Λ) + 1.

(11.10)

Usually, µ(Λ) is called the depth of the deepest holes of Λ. Clearly, we have µ(Λ) = maxn min {x, z} = max {o, x} . x∈E

z∈Λ

x∈D(o)

(11.11)

In this section we will introduce several results about r∗ (Sn ) by studying µ(Λ). Theorem 11.3 (Rogers [3]). r∗ (Sn ) < 2. Proof. More generally, we prove the following constructive assertion. Assertion 11.1. If Sn + Λ is a lattice sphere packing with µ(Λ) ≥ 3, then there exists a new lattice sphere packing Sn + Λ such that Λ ⊂ Λ and det(Λ ) = 13 det(Λ). It follows from (11.11) that there is a point x ∈ En such that x, z ≥ µ(Λ),

z ∈ Λ.

(11.12)

Also, it is clear that µ( 13 Λ) = 13 µ(Λ), and therefore there exists a lattice point u ∈ Λ such that x, 13 u ≤ 13 µ(Λ).

(11.13)

Clearly, 13 u ∈ Λ and thus det(Λ ) = 13 det(Λ), where Λ = Λ

1

3u

+Λ

2

3u

+Λ .

Then it follows from (11.12), (11.13), and the assumption of Assertion 11.1 that 13 u, z ≥ x, z − x, 13 u ≥ 23 µ(Λ) ≥ 2 holds for every point z ∈ Λ, and therefore u , v ≥ 2

11.2. Spherical Holes in Lattice Sphere Packings

177

for any distinct points u and v of Λ . In other words, Sn + Λ is a packing. Clearly, Theorem 11.3 is a consequence of (11.10) and Assertion 11.1. 2 Using a very complicated existence method, based upon the work of C.A. Rogers and C.L. Siegel, G.J. Butler was able to improve the previous theorem as follows. Theorem 11.4 (Butler [1]). r∗ (Sn ) ≤ 1 + o(1). Remark 11.1. Both Rogers’ original result and Butler’s improvement dealt not only with spheres, but also with general centrally symmetric convex bodies. By modifying Rogers’ constructive method for spheres, Henk [1] obtained √ r∗ (Sn ) ≤ 21/2 − 1. Remark 11.2. Analogously to Theorem 11.2 we have the follwing lower bound for r∗ (Sn ): 8 θ∗ (Sn ) ∗ r (Sn ) ≥ n ∗ − 1 0.51466. δ (Sn ) By studying diﬀerent patterns of the Dirichlet-Voronoi cells D(o) of ath proved the following the lattice sphere packings in E4 and E 5 , J. Horv´ result. Theorem 11.5 (Horv´ ath [2]). " √ √ 3−1 −1 r∗ (S4 ) = 2 3 8

and r∗ (S5 ) =

√ 3 13 + − 1. 2 6

The proofs of both Theorem 11.4 and Theorem 11.5 are very complicated. We omit them here. We list three more problems. Problem 11.3. Is it true that for all n ≥ 2, r∗ (Sn+1 ) ≥ r∗ (Sn )? Problem 11.4. Determine the value of lim r∗ (Sn ). n→∞

Problem 11.5. Is there a dimension n such that r∗ (Sn ) ≥ 1?

178

11. Holes in Sphere Packings

It is conjectured (see Conway and Sloane [1], Gruber and Lekkerkerker [1], and Rogers [14]) that the answer to Problem 11.5 is yes. If so, δ(Sn ) > δ ∗ (Sn ) holds in the corresponding dimension.

11.3. Cylindrical Holes in Lattice Sphere Packings In 1960, in a short note A. Heppes proved the following result: In every three-dimensional lattice sphere packing there is a straight line of inﬁnite length that does not intersect any of the spheres. In fact, his proof implied the existence of cylindrical holes (of inﬁnite length) in every threedimensional lattice sphere packing. Let ρ∗ (Sn ) be the maximum number such that in every n-dimensional lattice sphere packing Sn + Λ there is a cylindrical hole with an (n−1)-dimensional spherical base of radius ρ∗ (Sn ). Improving Heppes’ result, I. Hortob´ agyi determined the value of ρ∗ (S3 ). Also, J. Horv´ ath and S.S. Ry˘skov obtained a lower bound for ρ∗ (Sn ) in high dimensions. In this section we will introduce the main results of this kind. For convenience, we denote by H(x1 , x2 , . . . , xm ) the hyperplane determined by x1 , x2 , . . . , xm . A Seeber set of an n-dimensional lattice Λ is a set of n points {u1 , u2 , . . . , un } such that for i = 1, 2, . . . , n, ui is a point of Λ \ H(o, ui−1 , ui−2 , . . .) with minimal norm. We now introduce a basic lemma concerning the structure of a lattice. Lemma 11.5 (Horv´ ath [3]). Let Λ be a packing lattice of Sn and let {u1 , u2 , . . . , un } be a Seeber set of Λ. Denote by Hi the hyperplane {x : x, ui = 0} and by pi (x) the projection of x onto Hi . Then √ o, pi (u) ≥ 3 holds for every point u ∈ Λ \ H(o, ui ). Proof. First we proceed to show that for any point u ∈ Λ, o, ui ≤ max {o, u, u, ui} .

(11.14)

If u ∈ H(o, . . . , ui−1 ), then from the deﬁnition of a Seeber set it follows that o, ui ≤ o, u. If u ∈ H(o, . . . , ui−1 ), then u − ui ∈ H(o, . . . , ui−1 ) and hence o, ui ≤ u, ui .

11.3. Cylindrical Holes in Lattice Sphere Packings

179

These two cases imply (11.14).

ui u

o

pi (u)

Hi Figure 11.6 Now, assuming u ∈ Λ \ H(o, ui ), we consider the triangle ouui . It follows from (11.14) that oui is not the longest side of the triangle, and therefore (see Figure 11.6) max { oui u, uoui } ≥ π/3. On the other hand, since pi (u) = pi (u + kui ) for any integer k, we may assume that max { oui u, uoui } ≤ π/2. Thus, since o, u ≥ 2 and u, ui ≥ 2, o, pi(u) ≥ 2 sin (π/3) =

√ 3. 2

Lemma 11.5 is proved.

Although it is simple, this lemma plays an important role in the work of Heppes, Hortob´ agyi, Horv´ ath, and Ry˘skov on the cylindrical holes in lattice sphere packings. Theorem 11.6 (Heppes [2] and Hortob´ agyi [1]). ρ∗ (S3 ) =

√ 3 2 − 1. 4

In other words, every lattice sphere packing S3 + Λ has a cylindrical hole √ with a circular base of radius 3 2/4 − 1. Proof. Let Λ be a packing lattice of S3 , and let {u1 , u2 , u3 } be a Seeber set of √ Λ. We aim to show that if rS2 + p1 (Λ) is a covering in H1 , then r ≥ 3 2/4. In other words, letting T = v1 v2 v3 (where v1 = o) be a

180

11. Holes in Sphere Packings

triangle with vi ∈ p1 (Λ) and letting r(T ) be its circumradius, we proceed to show that √ 3 2 r(T ) ≥ . (11.15) 4 For convenience, we denote the circumcircle of T by S, and denote the cylinder with base S by Q.

u1 u2

u3 v2

o H1

v3 Figure 11.7

√ By Lemma 11.5, every side of T has√length not less than 3. Thus, simple computation yields that r(T ) ≥ 3 2/4 if one of its sides is greater than 2. So, without loss of generality, we may assume that o, u1 = 2, vi = p1 (ui ), i = 1, 2, 3, √ 3 ≤ vi , vj ≤ 2, i = j, and that the dihedral angle of ou1 u2 u3 at the edge ou3 is not greater than π/2. We deal with two cases (see Figure 11.7). Case 1. Both u2 and u3 lie on the same side of the plane H1 + 12 u1 . Without loss of generality, we may assume that o, u2 , and u3 lie on the same side of the plane H1 + 12 u1 and that u3 is the nearest to it. Then, we adjust the positions of u2 and u3 using the following process: Step 1. Replace u3 by its projection onto the plane H1 + 12 u1 . Step 2. Move u3 along the boundary of S + 12 u1 towards 12 u1 until u1 , u3 = o, u3 = 2. Step 3. Move u2 along the line parallel with ou1 until we have u2 , u3 = 2. Step 4. Move u2 along the curve bd(2S3 ) + u3 ∩ bd(Q) towards the

11.3. Cylindrical Holes in Lattice Sphere Packings

181

plane H1 + 12 u1 until we have o, u2 = u2 , u3 = 2.

√ Step 5. Move u2 along the circle {x : x − 12 u3 , u3 = 0, x, 12 u3 = 3} in the direction from u2 to u1 so that u1 , u2 = 2. Then we obtain a regular tetrahedron ou1 u2 u3 . Simple analysis shows that in this process the corresponding lattices are packing lattices of S3 and the corresponding circumradii r(T ) do not increase. Thus, (11.15) is true in this case. Case 2. The plane H1 + 12 u1 separates u2 and u3 . Without loss of generality, we may assume that o and u2 lie on the same side of H1 + 12 u1 . Then, we adjust the positions of u2 and u3 as follows: Step 1. Move u2 along the line parallel with ou1 towards H1 until o, u2 = 2. Then until u2 , u3 = 2. Step 2. Move u3 along the curve move u3 similarly bd(2S3 ) + u2 ∩ bd(Q) towards the plane H1 + 12 u1 so that u1 , u3 = 2. Then, we have o, u1 = o, u2 = u2 , u3 = u1 , u3 = 2, o, u3 > 2, and u1 , u3 > 2. Step 3. Move u2 along the circle {x : x− 12 u3 , u3 = 0, x, 12 u3 = c}, where c is a constant, in the direction from u2 to u1 so that u1 , u2 = 2. Then, adjust the position of u3 accordingly. At the end of this process we obtain a regular tetrahedron ou1 u2 u3 . Just as in Case 1, in this process the corresponding lattices are packing lattices of S3 and the corresponding circumradii r(T ) do not increase. Therefore, (11.15) is true in this case. Hence, Theorem 11.6 is proved.

2

Remark 11.3. By looking at the proof more closely, it seems that the extreme lattice sphere packing is unique up to isometry. Between ρ∗ (Sn ) and r∗ (Sn ) we have the following relationship, which provides a lower bound for ρ∗ (Sn ). Theorem 11.7 (Ry˘skov and Horv´ ath [1]). √ 3(1 + r∗ (Sn−1 )) ρ∗ (Sn ) ≥ − 1 0.3117. 2 Proof. Let Sn + Λ be a lattice sphere packing with the thinnest cylinder hole, and let {u1 , u2 , . . . , un } be a Seeber set of Λ. Then, it follows from Lemma 11.5 that √ ( 3/2)Sn−1 + pi (Λ) is a lattice sphere packing in Hi . From the deﬁnition of r∗ (Sn−1 ) we have √ 3(1 + r∗ (Sn−1 )) r≥ 2

182

11. Holes in Sphere Packings

if rSn−1 + pi (Λ) is a covering in Hi . On the other hand, we have pi (Sn + u) = Sn−1 + pi (u) for every point u ∈ Λ. Thus, using Remark 11.2, √ 3(1 + r∗ (Sn−1 )) ∗ ρ (Sn ) ≥ − 1 0.3117. 2 2

Theorem 11.7 is proved.

Remark 11.4. From the proof of Theorem 11.7 it follows that every lattice sphere√packing Sn + Λ has cylindrical holes, with spherical bases of radii at least 3(1 + r∗ (Sn−1 ))/2 − 1, in n independent directions. Analogous to Problems 11.2 and 11.4, we have the following problem concerning ρ∗ (Sn ). Problem 11.6. Determine the value of lim ρ∗ (Sn ). n→∞

12. Problems of Blocking Light Rays

12.1. Introduction We call a straight line with one end extending to inﬁnity a light ray. Let h(Sn ) be the smallest number such that there exists a ﬁnite sphere packing Sn + X, with X = {o, x1 , x2 , . . . , xh(Sn ) }, such that every light ray starting from o is blocked by one of the translates Sn + xi , xi ∈ X \ {o}. Similarly, when X is restricted to being a subset of a lattice, we denote the corresponding number by h∗ (Sn ). Hornich proposed the following problem (see L. Fejes T´oth [7]). Hornich’s Problem: Determine the numbers h(Sn ) and h∗ (Sn ). In 1959, L. Fejes T´ oth [7] obtained the ﬁrst result in this direction, proving h(S3 ) ≥ 19. He also proposed another related problem. Let (Sn ) be the smallest number such that there is a ﬁnite packing Sn + Y , where Y = {o, y1 , y2 , . . . , y (Sn ) }, and such that every light ray starting from any point of Sn is blocked by one of the translates Sn + yj , yj ∈ Y \ {o}. His problem can be formulated as follows. L. Fejes T´ oth’s Problem: Determine the number (Sn ). Remark 12.1. Let ∗ (Sn ) be the corresponding number in the lattice case. It follows from Theorem 11.6 and Theorem 11.7 that when n ≥ 3, ∗ (Sn ) = ∞.

184

12. Problems of Blocking Light Rays

It is obvious that

h(Sn ) ≤ h∗ (Sn ),

(12.1)

h(Sn ) ≤ (Sn ),

(12.2)

and, in particular, h(S2 ) = h∗ (S2 ) = (S2 ) = ∗ (S2 ) = 6. Also, relating k(Sn ) and h(Sn ) we have the following simple result. Theorem 12.1. If n ≥ 3, then k(Sn ) < h(Sn ). Proof. Let Sn , Sn +x1 , . . . , Sn +xk(Sn ) be an optimal kissing conﬁguration, and let Sn , Sn + y1 , . . . , Sn + yh(Sn ) be a ﬁnite packing such that every light ray starting from o is blocked by one of the translates Sn + yj , j = 1, 2, . . . , h(Sn ). For any point x, let Ω(x) be the intersection of bd(Sn ) and the smallest cone with vertex o, containing Sn +x. Clearly, Ω(x) is a cap on bd(Sn ). For convenience, we denote the surface area of a cap of geodesic radius π/6 by µn . From the deﬁnitions of k(Sn ) and h(Sn ) it follows that the caps Ω(xi ), i = 1, 2, . . . , k(Sn ), form a packing in bd(Sn ), the caps Ω(yj ), j = 1, 2, . . . , h(Sn ), form a covering in bd(Sn ), s(Ω(xi )) = µn , s(Ω(yj )) ≤ µn , and therefore

k(Sn )

k(Sn )µn =

s(Ω(xi )) < nωn

i=1

h(Sn )

<

s(Ω(yj ))

j=1

≤ h(Sn )µn . Theorem 12.1 follows.

2

Remark 12.2. Halberg, Levin, and Straus [1] and Boltjanski and Gohberg [1] studied an analogous problem by allowing overlap between the translates Sn + xi , xi ∈ X \ {o}. It follows from routine argument that this case is equivalent to a cap covering problem. Namely, what is the minimal number of caps of geodesic radii π/6 that can cover the surface of Sn ?

12.2. Hornich’s Problem

185

Remark 12.3. Substituting Sn by any centrally symmetric convex body C in Hornich’s problem and by any convex body K in L. Fejes T´ oth’s problem, one can deﬁne and study h(C), h∗ (C), (K), and ∗ (K). We call these numbers the Hornich number, lattice Hornich number, L. Fejes T´ oth number, and lattice L. Fejes T´ oth number of the corresponding convex body, respectively. Clearly, these numbers are invariant under linear transformations of the considered convex bodies.

12.2. Hornich’s Problem Clearly, determining the values of h(Sn ) and h∗ (Sn ) for n ≥ 3 is extraordinary diﬃcult. Perhaps, it is even more diﬃcult than determining the values of k(Sn ) and k∗ (Sn ). In E 3 , L. Fejes T´ oth’s lower bound for h(S3 ) was improved by Heppes [3] and Cs´ oka [1], while Danzer [1] obtained an upper bound. The known results about h(S3 ) can be expressed as 30 ≤ h(S3 ) ≤ 42. Since these results are still far from best possible and their methods are very technical, we will not go into the details. In this section we will obtain general bounds for h(Sn ) and h∗ (Sn ). First, let us introduce a geometric version of Dirichlet’s approximation theorem. Lemma 12.1 (Henk [2]). Let C be an n-dimensional centrally symmetric convex body centered at o, and let Λ be a lattice such that C +Λ is a packing. For any vector v = o there is a point u ∈ Λ and a positive integer j≤

2n δ(C) δ(C, Λ)

such that jv ∈ int(C) + u. Proof. Writing

6 m=

we consider the set X=

7 2n δ(C) , δ(C, Λ)

m

(Λ + iv) .

i=0

If 12 C + X is a packing, then δ( 12 C, X) = (m + 1)δ( 12 C, Λ)

186

12. Problems of Blocking Light Rays (m + 1)δ(C, Λ) 2n n 2 δ(C) δ(C, Λ) > δ(C, Λ) 2n = δ(C), =

which is impossible. Therefore, int 12 C + i1 v + u1 ∩ int 12 C + i2 v + u2 = ∅ holds for certain 0 ≤ i1 < i2 ≤ m, u1 ∈ Λ, and u2 ∈ Λ. In other words, (i2 − i1 )v ∈ int(C) + u1 − u2 . Thus, taking j = i2 − i1 and u = u1 − u2 , Lemma 12.1 is proved.

2

Theorem 12.2 (Zong [8]). 2

δ(C)n 2n (1+o(1)) h(C) ≤ h (C) ≤ , δ ∗ (C)n ∗

h(Sn ) ≤ h∗ (Sn ) ≤ 21.401n

2

(1+o(1))

.

Proof. It is well-known from convex geometry (see John [1] or Gruber and Lekkerkerker [1]) that for any n-dimensional centrally symmetric convex body C there is a linear transformation L such that √ Sn ⊆ L(C) ⊆ nSn . Thus, without loss of generality, we may assume that √ Sn ⊆ C ⊆ nSn .

(12.3)

Let C + Λ be a lattice packing with density δ∗ (C). By Lemma 12.1, for √ every vector v with v = n + 1 there is a positive integer j≤

2n δ(C) δ ∗ (C)

and a point u ∈ Λ \ {o} such that jv ∈ int(C) + u.

(12.4)

Let D be a Dirichlet-Voronoi cell associated with Λ. By Assertion 11.1, Remark 11.1, and (12.3) we have √ d(D) ≤ 3( n + 1). (12.5)

12.2. Hornich’s Problem Thus, writing

187 6

7 2n δ(C) m= , δ ∗ (C)

by (12.3), (12.4), and (12.5) we have √ h∗ (C) ≤ card m( n + 1)Sn ∩ Λ √ n ωn (m( n + 1) + d(D)) δ ∗ (C) ≤ v(C) √ √ n ωn (m( n + 1) + 3( n + 1)) δ ∗ (C) ≤ ωn 2

δ(C)n 2n (1+o(1)) ≤ . δ ∗ (C)n This proves the ﬁrst part of Theorem 12.2. Then, taking C = Sn , the second part follows from Theorem 3.1 and Theorem 8.2. 2 Remark 12.4. The original result of Zong [8], which is based on the number-theoretic version of Dirichlet’s approximation theorem, is weaker than Theorem 12.2. As a counterpart to Theorem 12.2 we have the following lower bound for h(Sn ). Theorem 12.3 (B´ ar´ any and Leader [1]). 2

h(Sn ) ≥ 20.275n

(1+o(1))

.

Proof. For convenience we write r = 20.275n , µ = 1/n, and ri = 2 + iµ. Let X be a subset of rSn such that Sn + X ∪ {o} is a ﬁnite packing, and write Xi = {x ∈ X : ri ≤ x < ri+1 } . Clearly we have Xi = ∅ if i > nr, and X=

nr

Xi .

(12.6)

i=0

Suppose that x ∈ Xi , and denote by ϕ/2 the spherical radius of the cap (Sn + x) ∩ bd(ri Sn ). Then (see Figure 12.1), for large n, 8 $ 2

2 2 ri + ri+1 −1 ϕ ϕ 2 sin = 1 − cos ≥ 1− 2 2 2ri ri+1 8

1 15 n − 4 ≥ . ri 16 n

188

12. Problems of Blocking Light Rays

Thus, by (8.1) and (8.21) we have card{Xi} ≤ m(n, ϕ/2) = m[n, ϕ] −n ≤ 20.599 sin(ϕ/2) ≤ rn 2−0.552n(1+o(1)).

(12.7)

Sn + x ri Sn

V (p)

Sn

Sn + p

Figure 12.1 Let p be an arbitrary point in E n . Let V (p) be the cone with vertex o over Sn + p, denote by Ω(p) the cap V (p) ∩ bd(Sn ), and let β be the spherical radius of Ω(p). Then, % s(Ω(p)) (n − 1)ωn−1 β ≤ (sin θ)n−2 dθ s(Sn ) nωn 0 ωn−1 (sin β)n−1 ≤ np1−n . ≤ nωn

(12.8)

By (12.6), (12.7), and (12.8) we have

nr s(Ω(x)) x∈X s(Ω(x)) = s(Sn ) nωn i=0 x∈Xi

≤ 2−0.552n(1+o(1))

nr

ri

i=0 2 −0.552n(1+o(1))

≤r 2 = 2−0.002n(1+o(1)).

This means that Sn + X can block at most half of the light rays starting from o when n is large. Therefore, considering the light rays not blocked by Sn + X and using (12.8) we have h(Sn ) ≥

2 rn−1 ≥ 20.275n (1+o(1)) . 2n

12.3. L. Fejes T´ oth’s Problem

189 2

Theorem 12.3 is proved. To end this section, we propose the following problem. Problem 12.1. Do there exist absolute constants c and c∗ such that 2

h(Sn ) = 2cn

(1+o(1))

or

h∗ (Sn ) = 2c

∗

n2 (1+o(1))

?

If so, determine them.

12.3. L. Fejes T´ oth’s Problem From the physical point of view, L. Fejes T´ oth’s problem is more natural, more fascinating, and more challenging than Hornich’s problem. In 1996, B¨ or¨ oczky and Soltan [1] proved that (Sn ) < ∞, which shows the fundamental diﬀerence between (Sn ) and ∗ (Sn ) (see Remark 12.1). Almost simultaneously, Zong [6] obtained the following upper bound for (Sn ) using a constructive method. Theorem 12.4 (Zong [6]). 2

(Sn ) ≤ (8e)n (n + 1)n−1 n(n

+n−2)/2

.

√ Proof. For convenience, we consider S = ( n/2)Sn instead of Sn and assume that n ≥ 3. Clearly, we have √ In ⊂ S ⊂ nIn . (12.9) Let α be a positive number strictly less than one, and take 9√ : k = αn + 1 .

(12.10)

Then, αIn ⊂ S ⊂ kαIn . Denote the intersections of αIn and kαIn with the hyperplane {x ∈ En : xi = 0} by J(i) and J (i), respectively. Let Xi = {x1 , x2 , . . . , xkn−1 } be a set of points such that J(i) + Xi is a tiling in J (i), and write √ Yi = xj + j nei : xj ∈ Xi , (12.11)

190

12. Problems of Blocking Light Rays

where ei indicates the ith unit basis vector. Then, it follows from (12.9) and (12.11) that S + Yi ∪ {o} is a ﬁnite packing. Let w be a point with wi > 0. We introduce another set of points √ wi + j n Zi (w) = (w + xj ) : xj ∈ Xi . (12.12) wi Let xj , yj , and zj be three geometrically corresponding points in Xi + w, Yi + w, and Zi (w), respectively. Then, zj is the intersection of the hyperplane {x ∈ E n : xi = yji } and the light ray of direction xj . By (12.9) and (12.12) it is clear that S + Zi (w) ∪ {o} is another ﬁnite packing. First, let us prove the following statement. Assertion 12.1. When w is a point with wi ≥ 3k n−1 n/2(1 − α), the spheres S+Zi(w) block all the light rays starting from S and passing through J (i) + w. Vj∗

αIn + w + xj

wj R

Vj

x

√

nIn Figure 12.2

Let w be a point with √ √ k n−1 n( n + α) wi ≥ , 1−α

(12.13)

and let xj be a point of Xi . Take √ wj =

α+

n √ (w + xj ) n

∗ and √ denote by Vj , Vj the two cones with the same vertex wj and over nIn , αIn + w + xj , respectively. Then (see Figure 12.2), we have

Vj∗ − wj = −Vj + wj

(12.14)

12.3. L. Fejes T´ oth’s Problem and Vj∗

191

β + √n √ (w + xj ) + βIn . = α+ n

(12.15)

β≥0

If R is a light ray starting from a point x ∈ Vj and passing through a point y ∈ Vj∗ , x = y, then the rest of the light ray is contained in Vj∗ . Otherwise, let H be the two-dimensional plane containing both R and wj . By considering the intersection H ∩ (Vj ∪ Vj∗ ) and applying the symmetry indicated by (12.14) one can easily obtain a contradiction. On the other hand, using (12.13) and since j ≤ kn−1 , it is easy to get β ≤ 1 when √ √ wi + j n β+ n √ = . α+ n wi √ Since J(i) ⊂ αIn , by (12.9) and (12.14) the light rays starting from nIn and passing through J(i) + w + xj can be blocked by the translate S+

√ wi + j n (w + xj ). wi

Then, Assertion 12.1 follows from the fact that J(i) + Xi + w is a tiling in J (i) + w.

F3

S

F2 F1

Figure 12.3 Write

P (ξ1 , ξ2 , . . . , ξn ) = x ∈ E n : |xi | ≤ 12 ξi , i = 1, 2, . . . , n

√ and denote its facet with xi = ξi /2 by Fi (ξ1 , ξ2 , . . . , ξn ). Take l = kα ≥ n and 6 n−1 7 3k n m=2 +1 . (12.16) 2(1 − α)

192

12. Problems of Blocking Light Rays

Abbreviate F1 (ml, ml, . . . , ml), F2 (2ml, (m + kn−1 )l, ml, . . . , ml), . . . , F (2ml, 2ml, . . . , 2ml, (m + kn−1)l) by F1 , F2 , . . . , Fn , respectively. Clearly, we have ml 3k n−1 n ≥ . (12.17) 2 2(1 − α) We proceed to prove the following assertion. Assertion 12.2. Every light ray starting from S intersects at least one of the 2n facets F1 , −F1 , F2 , −F2 , . . . , Fn , −Fn (see Figure 12.3). For convenience, we write Fi∗ = Fi (l, l, . . . , l). Then Fi blocks all the light rays starting from lIn and passing through a ﬁxed point p ∈ bd(mlIn ) if and only if Fi blocks all the light rays starting from Fi∗ and passing through p, in other words, as showed in Figure 12.4, if and only if

(m + kn−1 )l − 2pi l l ∗ Fi − ei + ei 2pi − l 2 2

(m + kn−1 − 1)l l + (12.18) p − ei ⊂ Fi . 2pi − l 2

mlIn

lIn p

Fi

Figure 12.4 Deﬁne qi reductively by

l m n−1 qn = m+k −1 +1 , 2 2m − 1 m + k n−1 − 1 l qi = qi+1 + m−1 2 for i = n − 1, n − 2, . . . , 2, and q1 = ml/2. Then, by (12.10), (12.16), and the assumption n ≥ 3, (m + kn−1 + 1)l ≤ qn ≤ qn−1 ≤ · · · ≤ q2 4

12.3. L. Fejes T´ oth’s Problem

193

n−2

m + k n−1 − 1 (n − 2)l qn + m−1 2

n−2

1 1 1 ml ≤ 1+ + 2n 2 3n 2

n 1 + 2/3n ml 1 1 ≤ 1+ 2 2n (1 + 1/2n)2 2 √ e ml ml ≤ < . 2 2 2

≤

(12.19)

From (12.18), (12.19), and routine computation based upon elementary geometry illustrated by Figure 12.4, it can be shown that every light ray starting from Fi∗ and passing through a point p = (p1 , p2 , . . . , pn ) ∈ bd(mlIn ) with l m + k n−1 − 1 pi ≥ max qj + = qi i+1≤j≤n m−1 2 and |pj | ≤ qj ,

j = i + 1, . . . , n,

can be blocked by Fi . On the other hand, it follows from (12.19) that for each point p ∈ bd(mlIn ) there is an index i such that |pi | ≥ qi

and |pj | ≤ qj

for j = i + 1, . . . , n. Therefore, by (12.9) all the light rays starting from S can be blocked by the union of the 2n facets F1 , −F1 , F2 , −F2 , . . . , Fn , −Fn . Assertion 12.2 follows. Let Wi = {wi,1, wi,2 , . . . , wi,τi } be a set of points such that J (i) + Wi is a tiling in Fi , where τi = 2i−1 mn−1 . (12.20) We deﬁne i =

Zi (w)

w∈Wi

and =

n i (−i ) . i=1

Based on the periodic distribution of Yi + Wi and the relationship between Yi + Wi and i , it is easy to see that int(S + v1 ) ∩ int(S + v2 ) = ∅ whenever v1 and v2 are diﬀerent points in i . Also, from the deﬁnition of Fi and by looking at the (j + t)th coordinates of vj and vj+t , it can be veriﬁed that int(S + vj ) ∩ int(S + vj+t ) = ∅

194

12. Problems of Blocking Light Rays

whenever vj ∈ j , vj+t ∈ j+t , and t is a positive integer. Therefore, S + ∪ {o} is a ﬁnite packing. On the other hand, by Assertion 12.1, Assertion 12.2, and (12.17) the spheres S + block all the light rays starting from S. Hence, by (12.10), (12.16), and (12.20), (Sn ) = (S) ≤ card{} = 2

n

card{i }

i=1

= 2k n−1

n i=1

τi = 2 (2n − 1) (km)n−1

3k n n ≤ 2 (2 − 1) +k 2(1 − α) (n+2)/2 n−1 n ≤ 8n αn (1 − α) n

n−1

n

holds for every number α with 0 < α < 1. Since the function g(α) = αn (1 − α) n 1 n n attains its maximum n+1 at α = n+1 , we have n+1

n2 2 1 (Sn ) ≤ 8 1 + (n + 1)n−1 n(n +n−2)/2 n n

2

≤ (8e)n (n + 1)n−1 n(n

+n−2)/2

. 2

Theorem 12.4 is proved.

Remark 12.5. As a generalization of John’s result, Leichtweiß [1] proved that for every n-dimensional convex body K there is a linear transformation L such that Sn ⊆ L(K) ⊆ nSn (for a proof of this result, we refer to Assertion 13.2). Applying Leichtweiß and John’s results, Zong [6] proved, in a similar manner to Theorem 12.4, that 2 (K) ≤ (8e)n (n + 1)n−1 n3(n −1)/2 for every n-dimensional convex body K, and that 2

(C) ≤ (8e)n (n + 1)n−1 nn

−1

for every n-dimensional centrally symmetric convex body C. Improving the upper bound for (C) with a nonconstructive method, we have the following theorem.

12.3. L. Fejes T´ oth’s Problem

195

Theorem 12.5. 2

(C) ≤ 48n

(1+o(1))

.

Remark 12.6. Using a modiﬁcation of Zong’s method, Talata [3] obtained a similar bound for general convex bodies. Meanwhile B´ ar´ any discovered a proof for the sphere case based on the ideas of Section 2. Our proof here is based on his method. In order to prove Theorem 12.5, as well as Henk’s lemma (Lemma 12.1), we require another basic result. Lemma 12.2 (Rogers and Zong [1]). Let K1 and K2 be two convex bodies in En . Then K2 can be covered by v(K1 − K2 )θ(K2 )/v(K1 ) translates of K1 . In particular, when K1 is centrally symmetric and K2 = γK1 , K2 can be covered by (1 + γ)n θ(K1 ) translates of K1 . Proof. For convenience, we assume that both K1 and K2 contain o. Let X be a set of points such that K1 + X is a covering in E n with covering density θ(K1 ), and let l be a large number. For any point y ∈ En we deﬁne f (y) = card {x ∈ X : (K2 + y) ∩ (K1 + x) = ∅} . It is easy to show that (K2 + y) ∩ (K1 + x) = ∅ if and only if y − x ∈ K1 − K2 . Thus, letting χ(y) be the characteristic function of K1 − K2 , we have % % f (y)dy = χ(y − x)dy lIn

x∈X

lIn

≤ v(K1 − K2 ) card {X ∩ (l + 2d(K1 )) In } v((l + 2d(K1 ))In )(θ(K1 ) + ) ≤ v(K1 − K2 ) . v(K1 ) Therefore,

6 f (y) ≤

7 v(K1 − K2 ) θ(K1 ) v(K1 )

holds for some point y ∈ En . The general case is proved. In the symmetric case, it is easy to see that K1 − K2 = (1 + γ)K1 . Thus, the assertion follows from the general case.

2

196

12. Problems of Blocking Light Rays

Proof of Theorem 12.5. Assume that C is centered at o, and let Λ be a lattice such that C + Λ is a packing with density δ∗ (C). For convenience, we write r = 14 , 6 7 δ(C)2n m= , rn δ ∗ (C) m = 3n θ(C) , ri = 14(m + i − 1)r, and L(i, j) = {u ∈ Λ : (2rC + u) ∩ bd(jri C) = ∅} . Clearly, we have 2rC + L(i, j) ⊂ (jri + 4r)C \ (jri − 4r)C.

(12.21)

By Lemma 12.1, for every point v ∈ bd(ri C) there is a j ≤ m and a lattice point u ∈ Λ such that jv ∈ rC + u and therefore (see Figure 12.5) rC + jv ⊂ 2rC + u. This means that all the light rays starting from rC can be blocked by m

(2rC + L(i, j)).

j=1

o

rC + u rC 2rC + u Figure 12.5

By Lemma 12.2, we assume that 2rC can be covered by m translates rC + x1 , rC + x2 , . . . , rC + xm . Then we have xi ∈ 3rC. Deﬁning L (i, j) =

jm + i − 1 u : u ∈ L(i, j) j(m + i − 1)

(12.22)

(12.23)

12.3. L. Fejes T´ oth’s Problem and

197

L (i, j) = L (i, j) + xi ,

(12.24)

it follows that all the light rays starting from rC + xi can be blocked by m

(2rC + L (i, j)).

j=1

Therefore, all the light rays starting from 2rC can be blocked by

m m

(2rC + L (i, j)).

(12.25)

i=1 j=1

Since C + Λ is a packing and 1 jm + i − 1 ≤ ≤ 1, 2 j(m + i − 1) 2rC + L (i, j) is a ﬁnite packing of 2rC. Also, by (12.21), (12.22), (12.23), and (12.24) it follows that 2rC + L (i, j) ⊂ (jr1 + 7(2i − 1)r)C \ (jr1 + 7(2i − 3)r)C. Denoting the set on the right-hand side of the above formula by C(i, j), it can be veriﬁed that the int(C(i, j)) are pairwise disjoint. Thus, the system (12.25) is a ﬁnite packing of 2rC in (m + 1)r1 C. Hence, by Theorem 3.1 and Remark 3.5, (C) ≤

v((m + 1)r1 C) (m + 1)n r1n ≤ v(2rC) 2n r n 2

≤ 48n

(1+o(1))

. 2

Theorem 12.5 is proved.

Based on (12.2), Theorem 12.3, and Theorem 12.5 we propose the following problems to end this section. Problem 12.2. Does there exist an absolute constant c such that 2

(Sn ) = 2cn

(1+o(1))

?

If so, determine it. Problem 12.3. Let Sn +X be a packing in En , let ς(Sn , X) be the maximal length of the segments contained in En \ {int(Sn ) + X}, and let ς(Sn ) = min ς(Sn , X). X

198

12. Problems of Blocking Light Rays

Does there exist a constant c such that ς(Sn ) = 2cn(1+o(1)) ?

12.4. L´ aszl´ o Fejes T´ oth L´ aszl´ o Fejes T´ oth was born in 1915 in Szeged, Hungary. In 1933 he entered the University of Budapest to study mathematics and physics, where L. Fej´er was one of his teachers. In 1938 he obtained a doctorate in mathematics. Soon afterwards, L. Fejes T´ oth served in the Hungarian army for two years. In 1941 he left the army and became an assistant at the University of Kolozsv´ ar. Four years later, he moved to Budapest and worked as a middle-school teacher there. At the same time he lectured on mathematics at the University of Budapest. In 1949 he became a professor of mathematics at the Technical University of Veszpr´em. In 1965, Professor L. Fejes T´ oth moved to the Mathematical Institute of the Hungarian Academy of Sciences, where he served as the director from 1970 to 1985. He retired in 1985. Professor L. Fejes T´ oth was a guest of many foreign universities such as the University of Freiburg, University of Wisconsin, University of Washington, Ohio State University, and the University of Salzburg. Professor L. Fejes T´ oth mainly worked in the areas of convex and discrete geometry, in which he published about 180 papers and two books. He has never had a Ph.D student. However, his work and open problems have inﬂuenced almost every contemporary discrete geometer, both domestic and foreign. Professor L. Fejes T´ oth is a member of the Hungarian Academy of Sciences and a corresponding member of the S¨ achsische Akademie der Wissenschaften zu Leipzig. He has been awarded honorary doctorates by the University of Salzburg and Veszpr´em University. In addition, he was honored by several Hungarian national prizes and a Gauss medal of the Braunschweigische Wissenschaftliche Gesellschaft.

13. Finite Sphere Packings

13.1. Introduction How can one pack m unit spheres in En such that the volume or the surface area of their convex hull is minimal ? Let m be a positive integer, and let u be a unit vector in En . Then, we deﬁne Lm = {2iu : i = 0, 1, . . . , m − 1} and Sm,n = conv {Sn + Lm } . Clearly, Sn + Lm is a packing in Sm,n . Writing µm (Sn ) =

mv(Sn ) , v(Sm,n )

it can be regarded as a local density of Sn + Lm in Sm,n . By routine computation it follows that µm (Sn ) =

mωn δ(Sn ). ωn + 2(m − 1)ωn−1

Based on this observation, in 1975 L. Fejes T´ oth [10] made the following conjecture about the volume case of the above problem. Sausage Conjecture: In En , n ≥ 5, the volume of the convex hull of m nonoverlapping unit spheres is at least ωn + 2(m − 1)ωn−1 .

200

13. Finite Sphere Packings

Also, equality is attained only if their centers are equally spaced on a line, a distance 2 apart. On the other hand, for the surface area case, Croft, Falconer, and Guy [1] formulated the following counterpart. Spherical Conjecture: When packing m unit spheres into a convex set so that the surface area is minimal, it seems likely that the optimum shape is roughly spherical if m is large. From an intuitive point of view, it is hard to imagine that both conjectures are correct. Nevertheless, the sausage conjecture was proved by Betke, Henk, and Wills [1] for suﬃciently large n, while the spherical conjecture was conﬁrmed independently by B¨ or¨ oczky Jr. [2] and Zong [1]. In this chapter we will deal mainly with the work of U. Betke, K. B¨ or¨ oczky Jr., M. Henk, J.M. Wills, and C. Zong.

13.2. The Spherical Conjecture The following theorem provides a positive solution for the spherical conjecture. ◦ Theorem 13.1 (B¨ or¨ oczky Jr. [2] and Zong [1]). Let Km,n be the family ◦ of convex bodies Km,n that contain m nonoverlapping n-dimensional unit spheres and at which the surface area attains its minimum. Write

ηm =

max

min

◦ ◦ rSn +x⊆Km,n ⊆r Sn +x Km,n ∈K◦ m,n x∈E n

r . r

Then lim ηm = 1.

m→∞

To prove this theorem, in addition to Theorem 1.2 and the isoperimetric inequality we need the following lemma. Lemma 13.1. For every n-dimensional convex body K there is a rectangular parallelepiped P such that P ⊆ K ⊆ n3/2 P. Proof. First, let us quote two fundamental results from Leichtweiß [1]. These results are not only useful in the proof of this theorem, but are also interesting and important.

13.2. The Spherical Conjecture

201

Assertion 13.1. Let α and β be two ﬁxed numbers with −1 ≤ α ≤ 1 and −1 ≤ β ≤ 1. If among all the ellipsoids

n 2 x ∈ E : λ(x1 − α)(x1 + β) + (xi ) ≤ 1 , n

Eλ =

i=1

where λ ≥ 0, the unit sphere takes the minimum volume, then αβ ≥ 1/n. It follows from routine computation that v(Eλ ) =

(α + β)2 2 1 + (1 + αβ)λ + λ 4

and therefore

n/2

(1 + λ)−(n+1)/2 ωn

* ∂(v(Eλ )) ** nαβ − 1 = ωn . ∂λ *λ=0 2

Thus, the assumption v(Eλ ) ≥ ωn for all λ ≥ 0 implies * ∂(v(Eλ )) ** ≥ 0, ∂λ *λ=0 and consequently αβ ≥

1 n.

Assertion 13.1 follows. Assertion 13.2. For every n-dimensional convex body K there exists an ellipsoid E such that 1 nE

⊆ K ⊆ E.

Assume that E is a minimal circumscribed ellipsoid of K; for convenience we take E to be centered at o. Let L be a linear transformation of E n such that L(E) = Sn . Then, Sn is a minimal circumscribed ellipsoid of L(K). If H1 = {x ∈ E n : x1 = α} and H2 = {x ∈ E n : x1 = −β} , where 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1 are two supporting hyperplanes of L(K), then every point x ∈ L(K) satisﬁes both n (xi )2 ≤ 1 i=1

and (x1 − α)(x1 + β) ≤ 0.

202

13. Finite Sphere Packings

Thus, it follows that L(K) ⊆ Eλ for all λ ≥ 0 and therefore ωn is the minimal value of v(Eλ ). Then, by Assertion 13.1 we have 1/n ≤ α ≤ 1, 1 n Sn

⊆ L(K) ⊆ Sn ,

and ﬁnally 1 nE

⊆ K ⊆ E.

This proves Assertion 13.2. By Assertion 13.2, if a1 , a2 , . . . , an are the n axes of E with lengths 1 , 2 , . . . , n , respectively, and if P = x ∈ E n : x, ai ≤ n−3/2 i , i = 1, 2, . . . , n , then we have P ⊆ n1 E ⊆ K ⊆ E ⊆ n3/2 P. 2

Lemma 13.1 is proved.

◦ Proof of Theorem 13.1. By Lemma 13.1, for every convex body Km,n there is a rectangular parallelepiped P such that ◦ P ⊆ Km,n ⊆ n3/2 P.

(13.1)

Let l1 , l2 , . . . , ln be the lengths of the n edges of P . It follows from (13.1) that n # ◦ mωn ≤ v(Km,n ) ≤ v(n3/2 P ) = n3n/2 li (13.2) i=1

and ◦ s(Km,n ) ≥ s(P ) =

n n # 2 li . l i=1 j=1 j

(13.3)

On the other hand, by Theorem 1.2, m unit spheres can be packed into a large sphere of radius $ m r=2n δ(Sn ) when m is suﬃciently large. Then, by (13.2), (13.3), and the minimum ◦ assumption of s(Km,n ) we have ◦ s(P ) ≤ s(Km,n ) ≤ s(rSn ),

13.2. The Spherical Conjecture

203

$

n−1 n n # 2 m li ≤ nωn 2 n l δ(Sn ) i=1 j=1 j ; 8
(n−1)/n n ≤ c1 (n) li . i=1

This implies

1/n # n n 1 li ≤ c2 (n) l i=1 i=1 i

and therefore

max1≤i≤n li ≤ c4 (n), c3 (n) ≤ min1≤i≤n li

(13.4)

where c1 (n), c2 (n), c3 (n), and c4 (n) are all positive numbers depending ◦ only on n. Thus, by (13.1) and (13.4), for every convex body Km,n there ◦ is a suitable number γ(Km,n ) such that ◦ ◦ c5 (n)Sn ⊆ γ(Km,n )Km,n ⊆ Sn ,

where c5 (n) is another constant depending only upon n. Clearly, ◦ lim γ(Km,n ) = 0.

m→∞

(13.5)

If, contrary to the assertion of Theorem 13.1, lim ηm = 1,

m→∞

then by Blaschke’s selection theorem, there is a nonspherical convex body ◦ ◦ K and a series of convex bodies Km , Km , . . . such that 1 ,n 2 ,n c5 (n)Sn ⊆ K ⊆ Sn and

◦ ◦ lim γ(Km )Km = K. j ,n j ,n

mj →∞

Then, by the isoperimetric inequality, there are three positive numbers r, α, and β such that v(rSn ) ≥ v(K) + α (13.6) and s(rSn ) ≤ s(K) − β.

(13.7)

Applying Theorem 1.2 once more, it follows from (13.5), (13.6), and (13.7) ◦ that (r/γ(Km ))Sn contains mj nonoverlapping unit spheres but with j ,n

204

13. Finite Sphere Packings

◦ ) when mj is suﬃciently large. This surface area smaller than s(Km j ,n contradiction proves lim ηm = 1. m→∞

2

Theorem 13.1 is proved.

Remark 13.1. In fact, Theorem 13.1 is true not only for the sphere and surface area but also for any n-dimensional convex body and any ﬁxed quermassintegral between 1 and n (see B¨ or¨ oczky Jr. [2], Zong [1], or Zong [4]).

13.3. The Sausage Conjecture For the sausage conjecture we have the following theorem. Theorem 13.2 (Betke, Henk, and Wills [1]). Let Km,n be an ndimensional convex body that contains m nonoverlapping unit spheres. When n is suﬃciently large, v(Km,n ) ≥ ωn + 2(m − 1)ωn−1 , where equality holds if and only if Km,n is congruent to Sm,n . Although the assertion of this theorem is clear and elegant, its proof is long and complicated. Let Sn + xi , i = 1, 2, . . . , m, be the m unit spheres packed in Km,n and write Xm,n = {x1 , x2 , . . . , xm } . As usual, we denote the Dirichlet-Voronoi cell with respect to xi by D(xi ). Clearly, we have m v(Km,n ) = v(D(xi ) ∩ Km,n ). (13.8) i=1

Thus, to prove Theorem 13.2 it is suﬃcient to show that v(D(xi ) ∩ Km,n ) ≥ 2ωn−1 for m − 2 points, v(D(xi ) ∩ Km,n ) ≥ ωn−1 + 12 ωn for 2 points, and that equality in these equations cannot hold simultaneously if Km,n is not congruent to Sm,n . For this purpose we consider a ﬁxed Dirichlet-Voronoi cell, say D = D(xm ) with respect to xm = o, and the corresponding convex body D∗ = D ∩ Km,n .

13.3. The Sausage Conjecture

205

Deﬁnition 13.1. Let ui be the unit vector in the direction of xi , and let φ=

max

1≤i, j≤m−1

{arccos(|ui , uj |)} .

Also, without loss of generality, let u1 and u2 be two vectors such that    φ, if φ ≥ π/3, φ, if ui , uj ≥ 0 for all indices, (13.9) arccos(|u1 , u2 |) =  arccos(|ui , uj |) , otherwise.  max

ui ,uj <0

Remark 13.2. In the third case of (13.9), writing Ω1 = x ∈ bd(Sn ) : u1 , x > 12 and Ω2 = {x ∈ bd(Sn ) : u1 , x ≤ u1 , u2 } , we have {u1 , u2 , . . . , um−1 } ⊂ Ω1 ∪ Ω2 , and the angle between any two vectors u ∈ Ω1 and v ∈ Ω2 is larger than π/3. By considering two subcases, routine argument shows that − cos(φ/2) ≤ u1 , u2 ≤ − cos φ.

(13.10)

Remark 13.3. Roughly speaking, a small φ indicates that near to o the convex hull, conv{Sn + Xm,n }, is “sausage-like” if u1 , u2 < 0 and is “needle-like” if u1 , u2 > 0. Meanwhile, since the sum of the three angles of any triangle is π, it follows that φ < π/3 and u1 , u2 > 0 cannot hold at three points of Xm,n . This fact plays a very important role in the proof of Theorem 13.2. Deﬁnition 13.2. Write G = conv{o, 2u1 , 2u2 }∩Sn , and denote by f (G, x) the map given by f (G, x) = y ∈ G, where x, y = min {x, w} . w∈G

Then, we deﬁne D1 = {x ∈ D∗ : f (G, x) ∈ rint(G)} , D2 = {x ∈ D∗ : f (G, x) ∈ (o, u1 ) ∪ (o, u2 )} , D3 = {x ∈ D∗ : f (G, x) = o} , and

D4 = {x ∈ D∗ : f (G, x) ∈ [2u1 , 2u2 ]} .

206

13. Finite Sphere Packings

It is clear that v(D∗ ) ≥

4

v(Di ).

(13.11)

i=1

By (13.8) and (13.11), the proof of Theorem 13.2 depends on complicated estimates of the values of v(Di ). Lemma 13.2. Let u and v be two orthogonal unit vectors, and let α and be two positive numbers such that α(α + ) ≥ 1 and (α + )v ∈ D∗ . Then [o, h1 (α, )u] + αv ∈ D∗ , where h1 (α, ) = / (α + )2 − 1. Proof. It follows from the assumption and the convexity of D∗ that conv {Sn , (α + )v} ⊂ D∗ .

(13.12)

u h1 (α, )u + αv

Sn

o

v

αv

(α + )v

Figure 13.1 With the help of Figure 13.1, since α ≥ 1/(α + ), routine argument yields h1 (α, )u + αv ∈ conv {Sn , (α + )v} , which implies [o, h1 (α, )u] + αv ⊂ conv {Sn , (α + )v} . Thus, Lemma 13.2 follows from (13.12) and (13.13). Lemma 13.3. v(G) ≥ φ/2. Proof. Write γ = u1 , u2 , θ = arccos(|γ|) and V = {αu1 + βu2 : α ≥ 0, β ≥ 0} .

(13.13) 2

13.3. The Sausage Conjecture

207

2u2 M u2

G

o

u1

2u1

Figure 13.2 Now, as illustrated in Figure 13.2, we deal with two cases. Case 1. γ ≥ − 12 . Then, we have θ = φ, V ∩ Sn ⊂ G, and therefore v(G) ≥ φ/2.

(13.14)

Case 2. γ < − 12 . Writing M = V ∩ Sn \ G, routine computation yields that v(G) = v(V ∩ Sn ) − v(M ) π−θ = − arccos x + x 1 − x2 , 2 where x = 2 sin(θ/2). Thus, π 3θ − arccos x − + x 1 − x2 2 2 x = arcsin x − 3 arcsin + x 1 − x2 2 ≥ 0.

v(G) − θ =

Then, by (13.10) it follows that v(G) ≥ θ ≥ φ/2. From (13.14) and (13.15) Lemma 13.3 is proved.

(13.15) 2

Before estimating the values of v(Di ) we introduce some new notation. H = {x = αu1 + βu2 : −∞ ≤ α, β ≤ +∞} , H ∗ = {x ∈ E n : u1 , x = u2 , x = 0} , Hi = {x ∈ E n : ui , x = 0} . We now proceed to estimate the values of v(Di ).

208

13. Finite Sphere Packings

Lemma 13.4. v(D1 ) ≥

h1 (1, cosec φ − 1)2 φ ωn−2 . 2

Proof. It follows from the deﬁnition of φ that |ui , uj | ≥ cos φ for i = 1, 2 and j = 1, 2, . . . , m − 1, which implies uj , v ≤ sin φ whenever v ∈ H ∗ ∩ bd(Sn ). Hence, from the deﬁnition of D∗ we obtain cosec φ (Hi ∩ Sn ) ⊂ D∗

(13.16)

for i = 1, 2, and cosec φ (H ∗ ∩ Sn ) ⊂ D∗ . Then, applying Lemma 13.2 with u ∈ H ∩ bd(Sn ),

v ∈ H ∗ ∩ bd(Sn ),

α = 1, and = cosec φ − 1, we have h1 (1, cosec φ − 1)G + H ∗ ∩ Sn ⊂ D1 and therefore, applying Lemma 13.3, v(D1 ) ≥

h1 (1, cosec φ − 1)2 φ ωn−2 . 2

Lemma 13.4 is proved.

2

Lemma 13.5. v(D2 ) ≥ h1 (1, cosec φ − 1)ωn−1 . Proof. In a similar way to the proof of Lemma 13.4, it follows from (13.16) and Lemma 13.2 that [o, h1 (1, cosec φ − 1)ui ] + Hi ∩ Sn ⊂ D∗

(13.17)

for i = 1, 2. For convenience, we denote the convex body on the left-hand side of (13.17) by Ji . Let v1 and v2 be unit vectors of H determined by ui , vi = 0 and ui , vj < 0 (see Figure 13.3), where {i, j} = {1, 2}. Then {x ∈ E n : vi , x ≥ 0, x ∈ int(Ji )} ⊆ D2 .

(13.18)

13.3. The Sausage Conjecture

209

u2 o u1

v2 v1 Figure 13.3 Hence, it follows from (13.18) and the deﬁnitions of Ji and D2 that v(D2 ) ≥ h1 (1, cosec φ − 1)ωn−1 . 2

Lemma 13.5 is proved. Lemma 13.6. If φ < π/3 and u1 , u2 > 0, then v(D3 ) ≥

π−φ ωn . 2π

Proof. Writing V ∗ = {x = αu1 + βu2 : α ≤ 0, β ≤ 0, x ∈ Sn } , routine argument shows that v(V ∗ ) = and

π−φ 2

(V ∗ + H ∗ ∩ Sn ) ∩ Sn ⊂ D3 .

Thus, we have v(D3 ) ≥

v(V ∗ ) π−φ v(Sn ) = ωn , v(S2 ) 2π 2

which proves our lemma. Lemma 13.7. If φ ≤ π/4 and u1 , u2 < 0, then v(D4 ) ≥

cos φ − sin φ ωn−1 . cos(φ/2)

Proof. Write u = (u1 − u2 )/u1 , u2 and H ∗∗ = {x ∈ E n : u, x = 0} .

210

13. Finite Sphere Packings

u1 o u2 uj

Figure 13.4 Since φ ≤ π/4 < π/3 it is easy to see that for 1 ≤ j ≤ m−1, u1 , uj ≥ cos φ if and only if u2 , uj ≤ − cos φ (see Figure 13.4). It follows that |u, uj | ≥

|u1 , uj − u2 , uj | ≥ cos φ, u1 , u2

which implies uj , v ≤ sin φ ∗∗

for any unit vector v ∈ H . Thus, for any λ ∈ [0, 1] we obtain 2λu1 + 2(1 − λ)u2 + v, uj 2λ(1 + cos φ) + 2 cos φ + sin φ, ≤ −2λ(1 + cos φ) + 2 + sin φ,

if u1 , uj ≥ cos φ, if u1 , uj ≤ − cos φ.

So, taking h2 (φ) =

1 + sin φ 2(1 + cos φ)

and keeping our assumption in mind, for λ ∈ [h2 (φ), 1 − h2 (φ)] we have 2λu1 + 2(1 − λ)u2 ∈ Sn , 2λu1 + 2(1 − λ)u2 + H ∗∗ ∩ Sn ⊂ D, and therefore

2λu1 + 2(1 − λ)u2 + H ∗∗ ∩ Sn ⊂ D∗ .

(13.19)

Let w be a unit vector determined by w = αu1 + βu2 for nonnegative α and β and satisfying u, w = 0. Also, let N = {x ∈ H ∗∗ ∩ Sn : w, x ≥ 0} . From (13.19) and the deﬁnition of D4 we have [2u1 , 2u2 ] + N ∩ D∗ ⊂ D4

13.3. The Sausage Conjecture

211

and therefore (1 − 2h2 (φ))2u1 , 2u2 ωn−1 2 cos φ − sin φ ≥ ωn−1 . cos(φ/2)

v(D4 ) ≥

2 √ Lemma 13.8. If φ > 0, then for every with 0 < < 2/ 3 − 1 we have Lemma 13.7 is proved.

v(D1 ) ≥

h1 (1, )2 φ ωn−2 , 2(1 + h3 (1 + , n))

√ where, for 1 ≤ β < 2/ 3, %

=%

1

(1 − x )

2 (n−5)/2

h3 (β, n) =

dx

1/β

(1 √ 3/2

1/β

− x2 )(n−5)/2 dx.

Proof. Using polar coordinates we have % % 1 v(D1 ) = dx dy. n − 2 G x∈H ∗ ∩bd(Sn ), y+x∈D∗

(13.20)

We now proceed to show that for a certain set G∗ ⊂ G with v(G∗ ) > 0, the above integral is of order ωn−2 . For this purpose we write Mρ = {x ∈ H ∗ ∩ bd(Sn ) : ρx ∈ D∗ } and

Mρ∗ = {x ∈ H ∗ ∩ bd(Sn ) : ρx ∈ D∗ } ,

and consider the inner integral at y = o. If Mρ = ∅, then H ∗ ∩ bd(ρSn ) intersects the aﬃne hull of certain facets Fj , say j = 1, 2, . . . , k, of the Dirichlet-Voronoi cell D. Let vj ∈ H ∗ be the outer unit normal of aﬀ{Fj }∩ H ∗ . It follows from a special case of Lemma 7.1 that aﬀ{Fj } ∩ (H ∗ ∩ bd(ρSn )) ⊂ int(Fj ∩ H ∗ )

√ for 1 ≤ ρ < 2/ 3 and there exists an αj ∈ [1, ρ] such that αj vj ∈ int(Fj ∩ H ∗ ). Writing

Mj = {x ∈ H ∗ ∩ bd(Sn ) : vj , x > αj /ρ} ,

we have Mρ =

k j=1

Mj .

212

13. Finite Sphere Packings

Deﬁning

√ Mj∗ = x ∈ H ∗ ∩ bd(Sn ) : 3αj /2 ≤ vj , x ≤ αj /ρ

and, for x ∈ Mj∗ ,

αj ≥ ρ, vj , x

γ(x) = we have

γ(x)x ∈ aﬀ{Fj } ∩ H ∗

and γ(x)x, αj vj 2 ≤

4 3

− α2j .

This implies γ(x)x ∈ Fj ∩ H ∗ , int(Mi∗ ) ∩ int(Mj∗ ) = ∅, and

k

i = j,

Mj∗ ⊂ Mρ∗ .

j=1

Hence, we have % dx = Mρ∗

+ Mρ∗

1+

+

dx +

Mρ

+

dx/

Mρ

+

dx

Mρ∗

dx

≥

(n − 2)ωn−2 + + 1 + Mj dx/ M ∗ dx

(13.21)

j

for a suitable index j ∈ {1, 2, . . . , k}. Let g(x) = (1 − x2 )(n−5)/2 , and for ξ ∈ [1, ρ] deﬁne %

1

f1 (ξ) =

g(x)dx ξ/ρ

and

% f2 (ξ) =

ξ/ρ

g(x)dx. √ 3ξ/2

Using polar coordinates and the monotonic property of f1 (ξ)/f2 (ξ) we have + dx f1 (αj ) f1 (1) + Mj = ≤ . f (α ) f2 (1) dx ∗ 2 j M j

On the other hand, it follows from (13.21) that % (n − 2)ωn−2 dx ≥ . 1 + h3 (ρ, n) ∗ Mρ

(13.22)

13.3. The Sausage Conjecture

213

Then, applying Lemma 13.2 with α = 1 and = ρ − 1 we obtain h1 (1, )G + Mρ∗ ⊆ D∗ .

(13.23)

√ Hence, for 0 < < 2/ 3 − 1, it follows from (13.20), (13.22), and (13.23) that h1 (1, )2 φ v(D1 ) ≥ ωn−2 . 2(1 + h3 (1 + , n)) 2

This proves Lemma 13.8.

Proof of Theorem 13.2. Using Lemmas 13.4–13.8, the proof is a consequence of the fact that ωn−1 lim = ∞. n→∞ ωn First we consider three cases depending on the value of φ and the sign of u1 , u2 . Case 1. 0 < φ < π/4 and u1 , u2 > 0. By Lemmas 13.4, 13.5, and 13.6 we have v(D∗ ) ≥ v(D1 ) + v(D2 ) + v(D3 )

ωn ωn (1 − sin(π/4))2 ≥ ωn−1 + ωn−2 − ωn−1 − +φ 2 2π 2(1 − sin2 (π/4)) ωn > ωn−1 + 2 for suﬃciently large n. Case 2. 0 < φ < π/4 and u1 , u2 < 0. By Lemmas 13.4, 13.5, 13.6, and the relation cos φ ≥ 1 − φ2 /2 we have v(D∗ ) ≥ v(D1 ) + v(D2 ) + v(D4 )

(1 − sin(π/4))2 πωn−1 ≥ 2ωn−1 + φ ω − 2ω − n−2 n−1 8 2(1 − sin2 (π/4)) > 2ωn−1 for suﬃciently large n. Case 3. φ ≥ π/4. Choose a suitable such that the assumption of Lemma 13.8 holds. Then, by Lemma 13.8 we have v(D∗ ) ≥ v(D1 ) ≥ > 2ωn−1 for suﬃciently large n.

h1 (1, )2 π ωn−2 8(1 + h3 (1 + , n))

214

13. Finite Sphere Packings

By Remark 13.2, Case 1 holds for at most two Dirichlet-Voronoi cells. Thus, when n is suﬃciently large and Km,n is not congruent to Sm,n , we have v(Km,n ) > ωn + 2(m − 1)ωn−1 . 2

Theorem 13.2 is proved. With explicit computation, Theorem 13.2 can be restated as follows.

Theorem 13.2∗ (Betke, Henk, and Wills [1]). When n ≥ 13387, L. Fejes T´ oth’s sausage conjecture is true. Remark 13.4. Using a more complicated method, considering a threedimensional section instead of a two-dimensional one, Theorem 13.2∗ was improved by Henk [1] and by Betke and Henk [1] to n ≥ 45 and n ≥ 42, respectively. Remark 13.5. Besides Betke, Henk, and Wills, many authors such as G. Fejes T´ oth, Gritzmann, Kleinschmidt, and Pachner have made contributions towards the sausage conjecture. For example, it was proved by Gritzmann [2] that v(Km,n ) >

ωn + 2(m − 1)ωn−1 √ 2+ 2

for every n-dimensional convex body Km,n that contains m nonoverlapping unit spheres.

13.4. The Sausage Catastrophe In lower dimensions, E 3 and E 4 , it seems that the ﬁnite sphere packing problem is much more complicated than its high-dimensional counterpart. In 1983, Wills [1] proposed the following phenomenon. The Sausage Catastrophe. In En , n = 3 or 4, the sausage arrangements are optimal when the number of spheres is small; then, for a certain number kn of spheres the extremal conﬁgurations become full-dimensional without going through intermediate-dimensional arrangements ﬁrst. In E 3 and E 4 it was proved by Betke, Gritzmann, and Wills [1] that intermediate-dimensional arrangements are not extremal. As for the values of k3 and k4 , it was conjectured by Wills [1] that k3 ≈ 56 and k4 ≈ 75000.

13.4. The Sausage Catastrophe

215

In this section we demostrate several examples to support this phenomenon. Example 13.1. Let K3,n be an n-dimensional convex body that contains three nonoverlapping unit spheres. Then v(K3,n ) ≥ ωn + 4ωn−1 , where equality holds if and only if K3,n is congruent to S3,n . Veriﬁcation. Suppose K3,n contains three nonoverlapping unit spheres Sn + v1 , Sn + v2 , and Sn + v3 , and v(K3,n ) is minimal. Without loss of generality, we may assume that v2 , v3 ≥ v1 , v2 = v1 , v3 = 2 and K3,n = conv {Sn + v1 , Sn + v2 , Sn + v3 } = Sn + T, where T is the triangle with vertices v1 , v2 , and v3 . Clearly, since Si is the largest section of Si+1 , we have ωi+1 < 2ωi . Then, writing θ = min { v1 v2 v3 , v1 v3 v2 } , we have 0 ≤ θ ≤ π/3 and v(K3,n ) = s(T )ωn−2 + 12 (4 + v2 , v3 )ωn−1 + ωn = 4 sin θ cos θ ωn−2 + 2(1 + cos θ)ωn−1 + ωn > 2(1 + cos θ + sin θ cos θ)ωn−1 + ωn . It is easy to see that f (x) = 1 + cos x + sin x cos x attains its minimum, 2, only at x = 0, for 0 ≤ x ≤ π/3. Thus, we have v(K3,n ) ≥ ωn + 4ωn−1 , and equality can be attained if and only if K3,n is congruent to S3,n . The example is veriﬁed. 2 Example 13.1 shows that the sausage arrangements are optimal for three spheres in E n , n ≥ 3. In E 3 , this result was extended to four spheres by B¨ or¨ oczky Jr. [1]. On the other hand, Gandini and Wills [1] showed that in E 3 for m = 56, 59–62, and all m ≥ 65 some full-dimensional packings of spheres are best possible and conjectured that in all remaining cases

216

13. Finite Sphere Packings

the sausage packing is optimal. The following example considers the case m = 56. Example 13.2 (Gandini and Wills [1]). There is a three-dimensional convex body K56,3 such that v(K56,3 ) < v(S56,3 ) = ω3 + 110ω2 . √ √ √ √ √ √ Veriﬁcation. Write u1 = ( 2, 2, 0), u2 = ( 2, 0, 2), u3 = (0, 2, 2), and deﬁne 3 Λ= zi ui : zi ∈ Z i=1

and Tk = conv {o, ku1 , ku2 , ku3 } . Then, S3 + Λ is a lattice packing and card {Tk ∩ Λ} =

k+3 . 3

T3

(13.24)

T6

T3

T2

T2

Figure 13.5 Let K be the convex polytope obtained by cutting oﬀ two translates of T2 and two translates of T3 from T6 (see Figure 13.5). Then, by (13.24) and routine computation we obtain card{K ∩ Λ} = 56. On the other hand, by routine computation we have v(S3 + K) < ω3 + 110ω2 = v(S56,3 ).

13.4. The Sausage Catastrophe

217

Thus, taking K56,3 = S3 + K, 2

Example 13.2 follows.

In E 4 , applying a similar method to a 24-cell with side length 34, we obtain the following result. Example 13.3 (Gandini and Zucco [1]). There is a four-dimensional convex body K375769,4 such that v(K375769,4 ) < v(S375769,4 ) = ω4 + 2 × 375768ω 3. Based on this result, Gandini and Zucco [1] gave the following modiﬁed version of Wills’ conjecture: k4 ≈ 375769.

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Index

Aﬄerbach, L. 40 Aigner, M. 21 automorphism 25 Babai, L. 40 Baire category 73 Ball, K. 54, 61 Bambah, R.P. 61 Bannai, E. 144, 146, 148 Baranovskii, E.P. 40, 45, 61 B´ ar´ any, I. 187, 195 Barlow, W. 19 Barnes, E.S. 40, 42, 45, 61 Bateman, P.T. 50 Bertrand, J.L.F. 43 Betke, U. 20, 200, 204, 214 Blaschke, W. 123 Blaschke’s selection theorem 203 Blichfeldt, H.F. 19, 40, 43, 91, 92, 94, 101, 103, 107, 156 blocking number 65, 71 Blundon, W.J. 162 Bolle, U. 156 Boltjanski, V.I. 184 Bonnet, P.O. 43

B¨ or¨ oczky, K. 111, 117, 165, 169, 189 B¨ or¨ oczky Jr, K. 110, 200, 204, 215 Bosanquet, L.S. 122 Bose-Chaudhuri-Hocquenghem code 81 Bouquet, J.C. 43 Bourgain, J. 77 Brunn-Minkowski inequality 44 Busemann, H. 123 Butler, G.J. 177 cap packing 11, 131 Cassels, J.W.S. 50, 62 cell complex 169 Chebyshev, P.L. 43 Christoﬀel-Darboux formula 126 closeness 165 closest sphere packing 165 cluster 16, 18 code 79, 83, 88, 143 codeword 79, 150 convex body 1, 65, 73 convex hull 199 Conway, J.H. 19, 84, 89, 142, 178 Cooke, R.G. 122 coordinate array 86

238 coorthosimplex 115, 117 Corput, J.G. van der 99 covering density 55, 57, 195 Coxeter, H.S.M. 58, 61, 111, 169 critical form 24, 27 Croft, H.T. 200 cross polytope 69, 74, 112 Cs´ oka, G. 185 cyclic code 80, 82 cylindrical hole 178, 181 Dalla, L. 66 Daniel’s asymptotic formula 107, 110 Danzer, L. 185 Dauenhauer, M.H. 19 Davenport, H. 19, 47, 50, 62, 99, 122 deepest hole 176 Dehn-Sommerville equation 70 Delone, B.N. 40, 43, 45, 61 Delone simplex 16, 61 Delone star 16, 18 Delone triangulation 16, 61 Delsarte, P. 127, 132, 145 densest lattice packing 4, 24, 29, 32, 35, 40, 72 densest translative packing 4 density 4, 13, 24, 32, 35, 40, 153, 157, 186 determinant 2 diﬀerence body 65 diﬀerence set 2 dihedral angle 107, 110, 121, 166, 172 Dirichlet, G.L. 43 Dirichlet’s approximation theorem 185 Dirichlet-Voronoi cell 5, 13, 15, 60, 103, 112, 119, 167, 170, 173, 186, 204, 211 discriminant 23

Index dodecahedron 13 Euler’s formula 8, 12, 173 extreme form 24, 33 face-centered cubic lattice 10, 17, 19 facet 34, 67, 69, 104, 168, 191, 211 Falconer, H.J. 200 Fej´er, L. 198 Fejes T´ oth, G. 163, 214 Fejes T´ oth, L. 14, 16, 19, 137, 165, 170, 183, 198, 199, 214 L. Fejes T´ oth number 185 L. Fejes T´ oth’s problem 183, 189 Ferguson, S.P. 18, 19 Few, L. 58, 61, 157 ﬁnite ﬁeld 79 F -module 80 fundamental domain 51, 53 fundamental parallelepiped 2 gamma function 2 Gandini, P.M. 215 Gauss, C.F. 19, 29, 43 generator matrix 80 generator polynomial 80, 82 geodesic arc 11, 16 geodesic distance 11 geodesic metric 111 geodesic radius 95, 97, 111, 184 Goethals, J.M. 132, 145 Gohberg, I. 184 Golay, M.J.E. 84 Golay code 81, 85 Green, R. 102 Gregory, D. 10 Gregory-Newton problem 10, 18, 21, 165 Gritzmann, P. 214 Groemer, H. 4, 19, 20, 128, 156 Gr¨ otschel, M. 142

Index Gruber, P.M. 20, 73, 178, 186 Gr¨ unbaum, B. 20 Guy, R.K. 200 Hadwiger, H. 2, 4, 20 Haj˘ os, G. 99 Halberg, C. 184 Hales, T.C. 14, 16, 18, 19 Hamming code 81, 84 Hamming distance 79, 82 Hardy, G.H. 100 Helmholtz, H. 43 Henk, M. 20, 110, 177, 185, 195, 200, 204, 214 Heppes, A. 162, 178, 185 Hermite, Ch. 43 Hermite’s constant 24, 40 Hermite’s reduction 40 Hilbert, D. 19, 44 Hilbert’s 18th problem 19 Hlawka, E. 4, 20, 47, 62, 99 h-locally averaged density 14 h-neighbor 14 homeomorphic 169 homogeneous harmonic polynomial 127 Hoppe, R. 11, 21 Hornich, H. 183 Hornich number 185 Hornich’s problem 183, 189 Hortob´ agyi, I. 178 Horv´ ath, J. 163, 175, 177, 179, 181 Hoylman, D.J. 20 Hsiang, W.Y. 14, 16, 19 Hua, L.K. 99 Hurwitz, A. 43 icosahedron 10 inner product distribution 143, 146 integral lattice 139, 144, 146 isomorphic 169 isoperimetric inequality 200

239 Jacobi polynomial 125, 141 Jensen’s inequality 7, 9, 171 John, F. 186, 194 Jordan, C. 44, 102 Jordan measurable set 51, 156 Justesen code 82, 88 Kabatjanski, G.A. 125, 133, 136 Kanagasahapathy, P. 157 Kepler, J. 13 Kepler’s conjecture 13, 14, 16, 18, 19, 165 Kershner, R. 61 k-fold packing 153, 155 Kirchhoﬀ, G.R. 43 Klein, F. 44, 102 Kleinschmidt, P. 214 Kneser, M. 140 Korkin, A.N. 33, 35, 38, 43 Kronecker, L. 43 Kronecker symbol 125 Kummer, E.E. 43 Kuperberg, W. 163 K-Z reduced form 36, 38, 39 K-Z reduction 40 Lagrange, J.L. 8, 19, 26, 43 Laplace operator 127 Larman, D.G. 20, 66, 74, 123 lattice 2, 9, 23, 36, 139, 153, 157, 183 lattice covering 55 lattice kissing number 2, 41 lattice packing 2, 7, 23, 42, 83 Leader, I. 187 Lebesgue measurable set 52 Lebesgue’s theorem 54 Leech, J. 11, 21, 82, 86, 88 Leech lattice 85, 140 Leichtweiß, K. 194, 200 Lekkerkerker, C.G. 20, 50, 178, 186 Lenstra, A.K. 40

240 Lenstra Jr, H.W. 40 Leven˘stein, V.I. 94, 125, 133, 136, 141 Levin, E. 184 Lie, S. 102 light ray 183, 188 Lindemann, F. 43 Lindsey II, J.H. 19 linear code 80, 83, 85, 88 Linhart, J. 163 Litsyn, S.N. 89 Littlewood, J.E. 100 Lloyd, S.P. 132 Lov´ asz, L. 40, 142 L-S-M reduced form 25, 27 L-S-M reduction 40 Macbeath, A.M. 50 Mahler, K. 19, 51, 62 Maly˘sev, A.V. 51, 62 Mani-Levitska, P. 66 Markov, A.A. 44 Milman, V.D. 77 Milnor, J. 19 Minkowski, H. 2, 20, 25, 43, 47 Minkowski-Hlawka theorem 44, 47, 50, 54, 84, 88 Minkowski inequality 74 Minkowski metric 44, 74 Minkowski sum 155 mixed volume 155 M¨ obius function 50 monomial 127 Mordell, L.J. 31 Muder, D.J. 19 Nemhauser, G.L. 142 Newton, I. 10, 21 Odlyzko, A.M. 132, 141 orthopoint 115 orthosimplex 114, 117, 119

Index Pachner, U. 214 pentagonal prism 18 perfect form 33, 35, 41 Petty, C.M. 123 P´ olya, G. 100 polyhedral convex cone 34 primitive point 50, 53 properly reduced form 51 quadratic form 19, 23, 25, 31, 34, 41, 47 quadratic residue code 81 quermassintegral 204 Rankin, R.A. 19, 94, 96, 99 Reed-Muller code 82, 85, 88 Reed-Solomon code 81 Riemann zeta function 45, 47 Riesz, F. 100 Rogers, C.A. 19, 47, 50, 54, 55, 58, 61, 103, 108, 122, 176, 178, 195 Rush, J.A. 84 Ry˘skov, S.S. 40, 45, 61, 175, 178, 181 Sanov, J.N. 51 saturated packing 14, 16, 17 sausage conjecture 199, 214 sausage catastrophe 214 Schaake, G. 99 Schl¨ aﬂi, L. 107 Schl¨ aﬂi’s function 107, 111, 121 Schmidt, W. 51, 54, 61 Schneider, T. 51, 62 Schrijver, A. 142 Sch¨ utte, K. 11, 21 score 16 Seeber, L.A. 19, 27 Seeber set 178, 181 Segre, B. 19 Seidel, J.J. 132, 145 Selling’s reduction 40

Index Shannon, C.E. 71 Shephard, G.C. 122 Sidelnikov, V.M. 94 Siegel, C.L. 51, 52, 62, 177 Siegel’s mean value formula 52, 54 simplical polytope 69 Sloane, N.J.A. 19, 82, 86, 88, 132, 137, 141, 142, 144, 146, 148, 178 Smith, H.J.S. 44 solid angle 58 Soltan, V. 189 space-centered cubic lattice 166, 175 spherical code 127, 145 spherical conjecture 200 spherical harmonic 128, 145 spherical l-design 145, 147 spherical polynomial 127 spherical simplex 107, 111, 121 stable form 24 star body 47 Steiner’s formula 155 Stirling’s formula 77, 110, 136 ˘ Stogrin, M.I. 40, 45 Straus, E. 184 sublattice 37 supersphere 74, 99 Swinnerton-Dyer, P. 72 Szeg¨ o, G. 125 Talata, I. 20, 77, 195 Tammela, P.P. 40, 45 Taylor’s theorem 96 Temesv´ ari, A.H. 163 thirteen spheres problem 10 Thue, A. 8, 19 tiling 6, 8, 51, 104, 154, 169 translative kissing number 2, 65, 72, 74, 111 translative packing 2 Tsfasman, M.A. 89

241 Venkov, B.A. 40, 45 Vet˘cinkin, N.M. 39 Voronoi, G.F. 34, 41, 43 Voronoi’s method 34 Waerden, B.L. van der 11, 21, 40 Watson, G.L. 42, 43, 45 Weber, H. 43 Weierstrass, C.T.W. 43 weight 79 Weil, A. 51, 62 Wills, J.M. 200, 204, 214, 215 Wirtinger, W. 62 Wolsey, L.A. 142 Wyner, J.M. 71 Yaglom, I.M. 137 Yakovlev, N.N. 162 Yudin, V.A. 95 Zassenhaus, H. 19 Ziegler, G. 21 Zolotarev, E.I. 33, 35, 38, 43 Zong, C. 20, 65, 67, 70, 72, 74, 186, 189, 194, 200, 204 Zucco, A. 217

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