On Generalized Kissing Numbers of Convex Bodies (II)

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 $κ_α^*(B^n)$ of the $n$-dimensional balls for $3\le n\le 8$ and $α=2\sqrt{3}-2$. Furthermore, the lattice kissing number of a four-dimensional cross-polytope is determined.

Figures (1)

• Figure 1: $||\hbox{$\frac{2}{3}$}{\bf v}_{i_1}+\hbox{$\frac{1}{3}$}{\bf v}_{i_2}||=2$ holds for $\{i_1,i_2\}\subset\{1,2,3\}$.

Theorems & Definitions (10)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof