On the kissing number of the cross-polytope

TL;DR

This paper establishes new bounds on the translative and lattice kissing numbers of the n-dimensional cross-polytope, improving previous bounds and identifying unique lattice configurations in four dimensions.

Contribution

It provides improved upper and lower bounds for the translative kissing number and characterizes the unique lattice configuration achieving the maximum in four dimensions.

Findings

New upper bound for translative kissing number: 2.9162^{(1+o(1))n}

Improved lower bound for kissing configurations: 1.1637^{(1-o(1))n}

Lattice kissing number in four dimensions: 40 for D_4^+ lattice

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.