Sphere Packings

256 Pages·1999·0.923 MB·English

by  Chuanming Zong, John Talbot

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,
GregoryandNewtondiscussedthekissingnumbersofspheresandproposed
the Gregory-Newtonproblem. Sincethen, theseproblemsandrelatedones
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.
Thereareseveralbooksdealingwithvariousaspectsofspherepackings.
For example, Conway and Sloane [1], Erdo¨s, Gruber, and Hammer [1], L.
Fejes To´th [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
holesin spherepackings,andfinitespherepackings. Notonly arethemain
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 reviewersfor 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
atSpringer-VerlaginNewYorkhavebeencourteous,competent,andhelp-
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.
Thiswork 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
En Euclidean n-dimensional space.
x A point (or a vector) of En with coordinates (x ,x ,...,x ).
1 2 n
o The origin of En.
(cid:1)x,y(cid:2) The inner product of two vectors x and y.
(cid:3)x,y(cid:3) The Euclidean distance between two points x and y.
(cid:3)x(cid:3) 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) The volume of K.
s(K) The surface area of K.
C Ann-dimensionalcentrallysymmetricconvexbodycenteredat
o.
(cid:3)x,y(cid:3) The Minkowski distance between x and y with respect to C.
C
S The n-dimensional unit sphere centered at o.
n
ω The volume of the n-dimensional unit sphere.
n
(cid:3)x,y(cid:3) The geodesic distance between two points x and y in bd(S ).
g n
I The n-dimensional unit cube {x: |x|≤ 1}.
n i 2
Z The set of all integers.