On the kissing number of the cross-polytope

Abstract

A new upper bound $\kappa_T(K_n)\leq 2.9162^{(1+o(1))n}$ for the translative kissing number of the $n$-dimensional cross-polytope $K_n$ is proved, improving on Hadwiger's bound $\kappa_T(K_n)\leq 3^n-1$ from 1957. Furthermore, it is shown that there exist kissing configurations satisfying $\kappa_T(K_n)\geq 1.1637^{(1-o(1))n}$, which improves on the previous best lower bound $ \kappa_T(K_n)\geq 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 $n\geq 1$, 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.