On the kissing number of the cross-polytope

TL;DR

The paper advances the translative kissing number problem for the cross-polytope by establishing a new asymptotic upper bound $κ_T(K_n) ≤ 2.9162^{(1+o(1))n}$ through a refined Blichfeldt enlargement approach, improving the general Hadwiger bound for this body. It also strengthens the known lower bound, proving $κ_T(K_n) ≥ 1.1637^{(1-o(1))n}$ with a combinatorial construction based on sets in $\{0,±1,±2\}^n$. In the lattice setting, the authors prove $κ_L(K_4)=40$, achieved uniquely by lattices $K_4$-equivalent to $D_4^+$, and analyze the deep-hole structure to show the corresponding covering radius equals 1. Together, these results sharpen the asymptotic picture of both translative and lattice kissing numbers for the cross-polytope and highlight the special role of the $D_4^+$ lattice in four dimensions.

Abstract

A new upper bound $κ_T(K_n)\leq 2.9162^{(1+o(1))n}$ for the translative kissing number of the $n$-dimensional cross-polytope $K_n$ is proved, improving on Hadwiger's bound $κ_T(K_n)\leq 3^n-1$ from 1957. Furthermore, it is shown that there exist kissing configurations satisfying $κ_T(K_n)\geq 1.1637^{(1-o(1))n}$, which improves on the previous best lower bound $ κ_T(K_n)\geq 1.1348^{(1-o(1))n}$ by Talata. It is also shown that the lattice kissing number satisfies $κ_L(K_n)< 12(2^n-1)$ 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 $κ_L(K_4)=40$ in four dimensions.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• proof
• Lemma 1
• proof
• Theorem 3
• proof
• proof : Proof of Theorem \ref{['thm:main0']}
• Theorem 4
• proof