On Generalized Kissing Numbers of Convex Bodies (II)

TL;DR

This paper determines exact generalized kissing numbers for n-dimensional balls (3 to 8) at a specific parameter and also finds the lattice kissing number for a 4D cross-polytope, advancing geometric understanding.

Contribution

It provides the first exact values of generalized kissing numbers for certain dimensions and a specific parameter, extending previous theoretical work.

Findings

Exact generalized kissing numbers for 3 to 8 dimensions at alpha=2√3−2

Lattice kissing number of a 4D cross-polytope determined

Advances understanding of geometric configurations in higher dimensions

Abstract

In 1694, Gregory and Newton discussed 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. Recently, Li and Zong introduced and studied the generalized kissing numbers of convex bodies. As a continuation of this project, in this paper we obtain the exact generalized kissing numbers $\kappa_\alpha^*(B^n)$ of the $n$-dimensional balls for $3\le n\le 8$ and $\alpha =2\sqrt{3}-2$. Furthermore, the lattice kissing number of a four-dimensional cross-polytope is determined.