Semidefinite programming bounds for the average kissing number

Abstract

The average kissing number of \(\mathbb{R}^n\) is the supremum of the average degrees of contact graphs of packings of finitely many balls (of any radii) in \(\mathbb{R}^n\). We provide an upper bound for the average kissing number based on semidefinite programming that improves previous bounds in dimensions \(3, \ldots, 9\). A very simple upper bound for the average kissing number is twice the kissing number; in dimensions \(6, \ldots, 9\) our new bound is the first to improve on this simple upper bound.