Semidefinite programming bounds for the average kissing number

TL;DR

This paper introduces semidefinite programming techniques to derive improved upper bounds for the average kissing number in dimensions 3 through 9, surpassing previous bounds and the simple twice-the-kissing-number estimate.

Contribution

The authors develop new semidefinite programming bounds that improve existing estimates for the average kissing number in certain dimensions.

Findings

Improved upper bounds for dimensions 3 to 9.

First bounds to beat the simple twice-the-kissing-number estimate in dimensions 6 to 9.

Semidefinite programming effectively tightens bounds on contact graph degrees.

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.