On the kissing number of the cross-polytope

Abstract

A new upper bound ${\kappa }_T(K_n)\le 2.9162^{(1+o(1))n}$ for the translative kissing number of the $n$-dimensional cross-polytope $K_n$ is proved, improving on Hadwiger's bound ${\kappa }_T(K_n)\le 3^n-1$ from 1957. Furthermore, it is shown that there exist kissing configurations satisfying ${\kappa }_T(K_n)\ge 1.1637^{(1-o(1))n}$, which improves on the previous best lower bound ${\kappa }_T(K_n)\ge 1.1348^{(1-o(1))n}$ by Talata. It is also shown that the lattice kissing number satisfies ${\kappa }_L(K_n)<12(2^n-1)$ for all $n\ge 1$, and that the lattice $D_4^{+}$ is the unique lattice, up to signed permutations of coordinates, attaining the maximum lattice kissing number ${\kappa }_L(K_4)=40$ in four dimensions.

10 pages