On Generalized Kissing Numbers of Convex Bodies

TL;DR

This paper introduces a generalized concept of kissing numbers for convex bodies, providing new exact results in dimensions three, four, and eight, advancing the mathematical understanding of sphere packings.

Contribution

It extends the classical kissing number problem to convex bodies and derives exact solutions in specific dimensions, which was not previously achieved.

Findings

Exact kissing numbers for convex bodies in dimensions three, four, and eight.

Generalization of the classical kissing number problem.

New mathematical techniques for analyzing convex body packings.

Abstract

In 1694, Gregory and Newton proposed the problem to determine the kissing number of a rigid material ball. This problem and its higher dimensional generalization have been studied by many mathematicians, including Minkowski, van der Waerden, Hadwiger, Swinnerton-Dyer, Watson, Levenshtein, Odlyzko, Sloane and Musin. In this paper, we introduce and study a further generalization of the kissing numbers for convex bodies and obtain some exact results, in particular for balls in dimensions three, four and eight.