SHARP SPHERE PACKINGS

Abstract

The classical sphere packing problem asks for the densest possible configuration of nonoverlapping equal balls in the three dimensional Euclidean space. This natural and even naive question remained open for several centuries and has driven a lot of research in geometry, combinatorics and optimization. The complete proof of the sphere packing problem was given by T. Hales in 1998 Hales [2005]. A similar question can be asked for Euclidean spaces of dimensions other then three or for spaces with other geometries, such as a sphere, a projective space, or the Hamming space. The packing problem is not only an exiting mathematical puzzle, it also plays a role in computer science and signal processing as a mathematical model of the error correcting codes. In this paper we will focus on the upper bounds for the sphere packing densities. There exist different methods for proving such bounds. One conceptually simple and still rather powerful approach is the linear programming. We are particularly interested in those packing problems, which can be completely solved by this method. We will call such arrangements of balls the sharp packings. The sharp packings have many interesting properties. In particular, the distribution of pairwise distances between the centers of sharply packed spheres gives rise to summation and interpolation formulas. In the last section of this paper we will discuss a new interpolation formula for the Schwartz functions on the real line.