Zong C., Talbot J. (eds.) Sphere Packings

New York: Springer, 1999. — 256 p.Sphere packings is one of the most fascinating and challenging subjects in mathematics. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with other subjects found. This book gives a full account of this fascinating subject, especially its local aspects, discrete aspects, and its proof methods. The book includes both classical and contemporary results and provides a full treatment of the subject.Basic NotationThe Gregory-Newton Problem and Kepler's ConjecturePackings of Circular Disks
The Gregory-Newton Problem
Kepler's Conjecture
L Fejes Tóth's Program and Hsiang's Approach
Delone Stars and Hales' Approach
Some General Remarks
Positive Definite Quadratic Forms and Lattice Sphere PackingsThe Lagrange-Seeber-Minkowski Reduction and a Theorem of Gauss
Mordell's Inequality on Hermite's Constants and a Theorem of Korkin and Zolotarev
Perfect Forms, Voronoi's Method, and a Theorem of Korkin and Zolotarev
The Korkin-Zolotarev Reduction and Theorems of Blichfeldt, Barnes, and Vetčinkin
Perfect Forms, the Lattice Kissing Numbers of Spheres, and Watson's Theorem
Three Mathematical Geniuses: Zolotarev, Minkowski, and Voronoi
Lower Bounds for the Packing Densities of Spheres
The Minkowski-Hlawka Theorem
Siegel's Mean Value Formula
Sphere Coverings and the Coxeter-Few-Rogers Lower Bound for δ(Sn)
Edmund Hlawka
Lower Bounds for the Blocking Numbers and the Kissing Numbers of Spheres
The Blocking Numbers of S and S
The Shannon-Wyner Lower Bound for Both b(Sn) and k(Sn)
A Theorem of Swinnerton-Dyer
A Lower Bound for the Translative Kissing Numbers of Superspheres
Sphere Packings Constructed from Codes
Codes
Construction A
Construction B
Construction C
Some General Remarks
Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres I
Blichfeldt's Upper Bound for the Packing Densities of Spheres
Rankin's Upper Bound for the Kissing Numbers of Spheres
An Upper Bound for the Packing Densities of Superspheres
Hans Frederik Blichfeldt
Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres II
Rogers' Upper Bound for the Packing Densities of Spheres
Schläfli's Function
The Coxeter-Böröczky Upper Bound for the Kissing Numbers of Spheres
Claude Ambrose Rogers
Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres III
Jacobi Polynomials
Delsarte's Lemma
The Kabatjanski-Levenstein Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres
The Kissing Numbers of Spheres in Eight and Twenty–Four Dimensions
Some Special Lattices
Two Theorems of Levenštein, Odlyzko, and Sloane
Two Principles of Linear Programming
Two Theorems of Bannai and Sloane
Multiple Sphere PackingsA Basic Theorem of Asymptotic Type
A Theorem of Few and Kanagasahapathy
Remarks on Multiple Circle Packings
Holes in Sphere Packings
Spherical Holes in Sphere Packings
Spherical Holes in Lattice Sphere Packings
Cylindrical Holes in Lattice Sphere Packings
Problems of Blocking Light RaysHornich's Problem
L Fejes Tóth's Problem
László Fejes Tóth
Finite Sphere PackingsThe Spherical Conjecture
The Sausage Conjecture
The Sausage Catastrophe