Sphere Packings pptx

Sphere Packings
Chuanming Zong
Springer
To Peter M. Gruber and David G. Larman
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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 re-
sults 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. Never-
theless, I assume sole responsibility for any remaining mistakes. The staff
at Springer-Verlag in New York have been courteous, competent, and help-
ful, 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 Hum-
boldt Foundation, and the National Scientific Foundation of China.
Berlin, 1999 C. Zong
Basic Notation
E
n
Euclidean n-dimensional space.
x A point (or a vector) of E
n
with coordinates (x
1
,x
2
, ,x

n
).
o The origin of E
n
.
x, y The inner product of two vectors x and y.
x, y The Euclidean distance between two points x and y.
x The Euclidean norm of x.
d(X) The diameter of a set X.
conv{X} The convex hull of X.
K An n-dimensional convex body.
int(K) The interior of K.
rint(K) The relative interior of K in the considered space.
bd(K) The boundary of K.
v(K)ThevolumeofK.
s(K) The surface area of K.
C An n-dimensional centrally symmetric convex body centered at
o.
x, y
C
The Minkowski distance between x and y with respect to C.
S
n
The n-dimensional unit sphere centered at o.
ω
n
Thevolumeofthen-dimensional unit sphere.
x, y
g
The geodesic distance between two points x and y in bd(S

n
).
I
n
The n-dimensional unit cube {x : |x
i
|≤
1
2
}.
Z The set of all integers.
x Basic Notation
Z
n
The n-dimensional integer lattice {z : z
i
∈ Z}.
ΛAnn-dimensional lattice.
det(Λ) The determinant of Λ.
Q(x) A positive definite quadratic form.
γ
n
The Hermite constant.
ℵ Acode.
x, y
H
The Hamming distance between x and y.
D(x) The Dirichlet-Voronoi cell at x.
k(S
n

) The kissing number of S
n
.
k
∗
(S
n
) The lattice kissing number of S
n
.
b(S
n
) The blocking number of S
n
.
δ(S
n
) The maximum packing density of S
n
.
δ
∗
(S
n
) The maximum lattice packing density of S
n
.
θ(S
n
) The minimum covering density of S

n
.
θ
∗
(S
n
) The minimum lattice covering density of S
n
.
r(S
n
) The maximum radius of a sphere that can be embedded into
every packing of S
n
.
r
∗
(S
n
) The maximum radius of a sphere that can be embedded into
every lattice packing of S
n
.
ρ
∗
(S
n
) The maximum radius of the spherical base of an infinite cylinder
that can be embedded into every lattice packing of S
n

.
h(S
n
) The Hornich number of S
n
.
h
∗
(S
n
) The lattice Hornich number of S
n
.
(S
n
) The L. Fejes T´oth number of S
n
.
Contents
Preface vii
Basic Notation ix
1. The Gregory–Newton Problem
and Kepler’s Conjecture 1
1.1. Introduction 1
1.2. Packings of Circular Disks 7
1.3. The Gregory-Newton Problem 10
1.4. Kepler’s Conjecture 13
1.5. Some General Remarks 18
2. Positive Definite Quadratic Forms
and Lattice Sphere Packings 23

2.1. Introduction 23
2.2. The Lagrange-Seeber-Minkowski Reduction
and aTheoremof Gauss 25
2.3. Mordell’s Inequality on Hermite’s Constants
and a Theorem of Korkin and Zolotarev . . . . . . . . . . . . . . . . 31
2.4. Perfect Forms, Voronoi’s Method, and a Theorem
of Korkin and Zolotarev . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5. The Korkin-Zolotarev Reduction and Theorems
of Blichfeldt, Barnes, and Vet˘cinkin 36
2.6. Perfect Forms, the Lattice Kissing Numbers
of Spheres, and Watson’s Theorem . . . . . . . . . . . . . . . . . . . 41
xii Contents
2.7. Three Mathematical Geniuses: Zolotarev,
Minkowski, andVoronoi 42
3. Lower Bounds for the Packing Densities of Spheres 47
3.1. The Minkowski-Hlawka Theorem 47
3.2. Siegel’s Mean Value Formula 51
3.3. Sphere Coverings and the Coxeter-Few-Rogers
Lower Bound for δ(S
n
) 55
3.4. Edmund Hlawka 62
4. Lower Bounds for the Blocking Numbers
and the Kissing Numbers of Spheres 65
4.1. The Blocking Numbers of S
3
and S
4
65
4.2. The Shannon-Wyner Lower Bound for Both

b(S
n
)andk(S
n
) 71
4.3. A Theorem of Swinnerton-Dyer 72
4.4. A Lower Bound for the Translative Kissing Numbers
of Superspheres 74
5. Sphere Packings Constructed from Codes 79
5.1. Codes 79
5.2. Construction A 82
5.3. Construction B 84
5.4. Construction C 85
5.5. Some General Remarks 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 103
7.1. Rogers’ Upper Bound for the Packing Densities
of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2. Schl¨afli’s Function 107
7.3. The Coxeter-B¨or¨oczky Upper Bound
for the Kissing Numbers of Spheres . . . . . . . . . . . . . . . . 111

7.4. Claude Ambrose Rogers . . . . . . . . . . . . . . . . . . . . . . . . 122
Contents xiii
8. Upper Bounds for the Packing Densities
and the Kissing Numbers of Spheres III 125
8.1. Jacobi Polynomials 125
8.2. Delsarte’s Lemma 127
8.3. The Kabatjanski-Leven˘stein Upper Bounds
for the Packing Densities and the Kissing Numbers
of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9. The Kissing Numbers of Spheres in Eight
and Twenty–Four Dimensions 139
9.1. Some Special Lattices . . . . . . . . . . . . . . . . . . . . 139
9.2. Two Theorems of Leven˘stein, Odlyzko, and Sloane . . . . . 141
9.3. Two Principlesof Linear Programming 142
9.4. Two Theoremsof Bannai and Sloane 143
10. Multiple Sphere Packings 153
10.1. Introduction 153
10.2. A Basic Theorem of Asymptotic Type . . . . . . . . . . . . . 154
10.3. A Theorem of Few and Kanagasahapathy . . . . . . . . . . . . . 157
10.4. Remarkson Multiple Circle Packings 162
11. Holes in Sphere Packings 165
11.1. Spherical Holes in Sphere Packings . . . . . . . . . . . . . . . . 165
11.2. Spherical Holes in Lattice Sphere Packings . . . . . . . . . . 176
11.3. Cylindrical Holes in Lattice Sphere Packings . . . . . . 178
12. Problems of Blocking Light Rays 183
12.1. Introduction 183
12.2. Hornich’s Problem 185
12.3. L. Fejes T´oth’s Problem 189
12.4. L´aszl´o Fejes T´oth 198
13. Finite Sphere Packings 199

13.1. Introduction 199
13.2. The Spherical Conjecture . . . . . . . . . . . . . . . . . . . . . 200
13.3. The Sausage Conjecture 204
13.4. The Sausage Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . 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 E
n
. 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
S
n
=

x =(x
1
,x
2
, ,x
n

):
n

i=1
(x
i
)
2
≤ 1

and the n-dimensional unit cube
I
n
=

x =(x
1
,x
2
, ,x
n
): max
1≤i≤n
|x
i
|≤
1
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 S
n
by ω
n
,it
is well-known that
ω
n
=
π
n/2
Γ(n/2+1)
,
and therefore
s(S
n
)=nω
n
=
nπ
n/2
Γ(n/2+1)
,
2 1. The Gregory-Newton Problem, Kepler’s Conjecture
where Γ(x)isthegamma function.
If a
i
=(a
i1

,a
i2
, ,a
in
), i =1,2, , n,aren linearly independent
vectors in E
n
, then the set
Λ=

n

i=1
z
i
a
i
: z
i
∈ Z

,
where Z is the set of all integers, is called a lattice,andwecall{a
1
, a
2
, ,
a
n
} a basis for Λ. As usual, the absolute value of the determinant |a

ij
| is
called the determinant of the lattice and is denoted by det(Λ). Geometri-
cally speaking, det(Λ) is the volume of the fundamental parallelepiped
P =

n

i=1
λ
i
a
i
:0≤ λ
i
≤ 1

of Λ.
Let X be a set of discrete points in E
n
. We shall call K +X a translative
packing of K if
(int(K)+x
1
) ∩ (int(K)+x
2
)=∅
whenever x
1
and x

2
are distinct points of X. In particular, we shall call it
a lattice packing of K when X is a lattice.
Let k(K)andk
∗
(K) denote the translative kissing number and the lat-
tice kissing number of K, respectively. In other words, k(K)isthemax-
imum 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)andk(K).
Theorem 1.1 (Minkowski [7] and Hadwiger [1]). Let K be an n-
dimensional convex body. Then
k
∗
(K) ≤ k(K) ≤ 3
n
−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 + x
1
) ∩ (A + x
2
) = ∅
if and only if

1
2
D(A)+x
1

∩

1
2
D(A)+x
2

= ∅.
Therefore, by considering the two cases A =int(K)andA = 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,wehave

3y ∈ 3K
and
3
2
(z − y) ∈ 3K.
Hence we have
z =
1
3
×3y +
2
3
×
3
2
(z − y) ∈ 3K,
and therefore (see Figure 1.1)
K + x ⊆ 3K.
KK+ x
3K
Figure 1.1
Let K +x
i
,i=1, 2, ,k(K), be nonoverlapping translates that touch
K at its boundary. Then
k(K)

i=0
(K + x
i

) ⊆ 3K,
4 1. The Gregory-Newton Problem, Kepler’s Conjecture
where x
0
= o, and therefore
(k(K)+1)v(K) ≤ v(3K)=3
n
v(K),
which implies that
k(K) ≤ 3
n
−1,
and hence using (1.1), Theorem 1.1 is proved. ✷
Remark 1.1. In fact, n-dimensional parallelepipeds are the only convex
bodies in E
n
such that
k
∗
(K)=k(K)=3
n
−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 lI
n
. We define
δ(K) = lim sup
l→∞
m(K, l)v(K)

v(lI
n
)
,
the density of the densest translative packings of K in E
n
. Similarly, the
density δ
∗
(K)ofthedensest lattice packings of K is defined by restricting
the translative vectors to those in a lattice. In this case, it can be deduced
that
δ
∗
(K)=sup
Λ
v(K)
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 ∩lI
n
}v(K)

v(lI
n
)
.
Then simple analysis yields that
δ(K)=sup
X
δ(K, 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)
v(lA)
= δ(K).
1.1. Introduction 5
Proof. Let m
∗
(K, l) be the maximum number of nonoverlapping translates
K + x that intersect lI
n
and let d be the diameter of K.Since
lim
l→∞
v((l +2d)I
n

) − v((l −2d)I
n
)
v(lI
n
)
=0, (1.3)
from the definition of δ(K) it follows that for each >0thereisat

such
that
m(K, t

)v(K)
v(t

I
n
)
>δ(K) − 
and
m
∗
(K, t

)v(K)
v(t

I
n

)
<δ(K)+.
On the other hand, for any convex body A, there exists an l

> 0such
that for each l>l

there are two families of nonoverlapping cubes,

t

l
I
n
+ x
i
: i =1, 2, ,g

and

t

l
I
n
+ y
i
: i =1, 2, ,h

,

such that
g

i=1

t

l
I
n
+ x
i

⊆ A,
v(A) ≤
v

∪
g
i=1

t

l
I
n
+ x
i

1 −

,
A ⊆
h

i=1

t

l
I
n
+ y
i

,
and
v(A) ≥
v

∪
h
i=1

t

l
I
n
+ y
i


1+
.
Hence, when l>l

, one has
(1 −)(δ(K) − ) <
m(K, lA)v(K)
v(lA)
< (1 + )(δ(K)+),
which implies the assertion of Theorem 1.2. ✷
Remark 1.2. Clearly, k(K), k
∗
(K), δ(K), and δ
∗
(K) are invariant under
nonsingular linear transformations.
Suppose that X = {x
1
, x
2
, } is a discrete set of points in E
n
.For
each point x
i
∈ X the Dirichlet-Voronoi cell D(x
i
) is defined by
D(x

i
)={x : x, x
i
≤x, x
j
 for all x
j
∈ 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(x
i
)=

x
j
∈X

x : x − x
i
, x
j
−x
i
≤
1
2
x
j

+ x
i
, x
j
−x
i


,
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
E
n
=

x
i
∈X
D(x
i
),
and
int(D(x
j
)) ∩ int(D(x
k
)) = ∅
for every pair of distinct points x
j

and x
k
of X. In other words, the cells
D(x
i
)formatiling of E
n
.Thus,ifK + X is a packing, D(x
i
)canbe
regarded as a local cell associated to K + x
i
. In particular, when K is a
sphere centered at o,then
K + x
i
⊂ D(x
i
)
for every point x
i
∈ 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(x
i
)) <c
for every point x
i
∈ X,and

lim
l→∞
card{X ∩lI
n
}v(K)
v(lI
n
)
≥ δ(K) −.
Then, by (1.3), one has
δ(K) ≤
v(K)
λ(X, l)
+2 (1.4)
for sufficiently large l,where
λ(X, l)=

x
i
∈X∩lI
n
v(D(x
i
))
card{X ∩ lI
n
}
.
In particular, since all the Dirichlet-Voronoi cells are congruent if X is a
lattice, one has

δ
∗
(K)=sup
Λ
v(K)
v(D(o))
, (1.5)
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 S
2
.
Then,
v(P ) ≥ p tan
π
p
,
where equality holds if and only if P is regular and its edges are tangent to
bd(S
2
).
Proof. Without loss of generality, we may assume that the p edges L
1
, L
2

,
, L
p
of P are tangent to bd(S
2
)att
1
, t
2
, , t
p
in a circular order (see
Figure 1.2). Letting θ
i
be the angle between t
i
and t
i+1
,wheret
p+1
= t
1
,
it follows that
p

i=1
θ
i
=2π

and
v(P )=
p

i=1
tan
θ
i
2
.
S
2
t
1
t
2
t
3
θ
1
θ
2
Figure 1.2
Since the function f(x) = tan x is strictly concave on [0,π/2), by Jensen’s
inequality it follows that
v(P ) ≥ p tan

p
i=1
θ

i
2p
= p tan
π
p
,
where equality holds if and only if P is a regular polygon. Lemma 1.1 is
proved. ✷
Let Λ
2
be the two-dimensional lattice with basis {(2, 0), (1,
√
3)}.Then
S
2
+Λ
2
is a lattice packing of S
2
. From this example it follows that
k
∗
(S
2
) ≥ 6(1.6)
8 1. The Gregory-Newton Problem, Kepler’s Conjecture
and
δ
∗
(S

2
) ≥
v(S
2
)
det(Λ
2
)
=
π
√
12
. (1.7)
On the other hand, if S
2
+x
1
and S
2
+x
2
are two nonoverlapping circular
disks that touch S
2
at its boundary, then the angle between x
1
and x
2
is
at least π/3. Therefore,

k(S
2
) ≤
2π
π/3
=6. (1.8)
From (1.1), (1.6), and (1.8) it follows that
k
∗
(S
2
)=k(S
2
)=6.
The following result determines the exact values of δ
∗
(S
2
)andδ(S
2
).
Theorem 1.3 (Lagrange [1] and Thue [2]).
δ
∗
(S
2
)=δ(S
2
)=
π

√
12
.
Proof. Let X be a set such that S
2
+ X is a packing and
d(D(x)) ≤ c
for every point x ∈ X,wherec is a suitable positive number. By (1.4)
δ(S
2
) ≤
v(S
2
)
λ(X, l)
+2 (1.9)
for sufficiently large l,where
λ(X, l)=

x∈X∩lI
2
v(D(x))
card{X ∩ lI
2
}
. (1.10)
For convenience, we write
X
∗
= X ∩ (l +2c)I

2
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 ∩lI
2
, (1.11)
and the D
∗
(x) form a tiling of E
2
.Lete, f,andv 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)andq(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 ≤


v∈V
q(v)=2e. (1.13)
Then from (1.12) and (1.13) it follows that
e +6≤ 3f. (1.14)
Writing
p(X, l)=

x∈X∩lI
2
p(x)
card{X ∩lI
2
}
,
(1.3), (1.11), and (1.14) imply that
p(X, l) ≤
2e
f
+  ≤ 6 −
12
f
+ 
for sufficiently large l, and hence
lim sup
l→∞
p(X, l) ≤ 6. (1.15)
By (1.10) and Lemma 1.1 it follows that
λ(X, l) ≥
π


x∈X∩lI
2
p(x)
π
tan
π
p(x)
card{X ∩ lI
2
}
.
Since the function
f(x)=
x
π
tan
π
x
is concave when x ≥ 3, by Jensen’s inequality and (1.15) it follows that
λ(X, l) ≥ π
p(X, l)
π
tan
π
p(X, l)
≥
√
12.
Hence by (1.9) one has

δ(S
2
) ≤
π
√
12
. (1.16)
Then Theorem 1.3 follows from (1.2), (1.7), and (1.16). ✷
Remark 1.3. From the proof of Theorem 1.3 it follows that the lattice with
which
δ
∗
(S
2
)=
π
√
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 a
1
=(2, 0, 0), a
2
=(1,
√
3, 0), and a
3
=(1,

√
3/3, 2
√
6/3), and let Λ
3
be the lattice generated by them. Usually, Λ
3
is known as the face-centered
cubic lattice. It is easy to see that S
3
+Λ
3
is a packing of S
3
,inwhich
every sphere touches 12 others. This observation implies
k
∗
(S
3
) ≥ 12. (1.17)
In 1694, during a famous conversation, D. Gregory and I. Newton dis-
cussed 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 S
3

+Λ
3
, locally speaking, the kissing configuration is stable. In other
words, none of the twelve spheres that touch S
3
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 S
3
+ x
i
, i =1, 2, , k(S
3
), be nonoverlapping spheres that touch
S
3
at its boundary, and define
Ω
i
=

x ∈ bd(S
3
): x, x
i
≥

√
3

.