Sphere packings revisited

Abstract

In this paper we survey most of the recent and often surprising results on packings of congruent spheres in d -dimensional spaces of constant curvature. The topics discussed are as follows: – Hadwiger numbers of convex bodies and kissing numbers of spheres; – touching numbers of convex bodies; – Newton numbers of convex bodies; – one-sided Hadwiger and kissing numbers; – contact graphs of finite packings and the combinatorial Kepler problem; – isoperimetric problems for Voronoi cells, the strong dodecahedral conjecture and the truncated octahedral conjecture; – the strong Kepler conjecture; – bounds on the density of sphere packings in higher dimensions; – solidity and uniform stability. Each topic is discussed in details along with some of the “most wanted” research problems.