Upper bounds for packings of spheres of several radii

TL;DR

This paper develops new upper bounds for sphere packings of multiple radii using semidefinite programming, extending existing linear bounds and providing improved results for various packing problems.

Contribution

It introduces semidefinite programming techniques to extend linear bounds for packings of different shapes and sizes, offering new and improved bounds.

Findings

New bounds for packings of two different spherical cap sizes

Improved bounds for binary sphere packings

Slight enhancement of classical sphere packing bounds

Abstract

We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve bounds for the classical problem of packing identical spheres.