New covering and illumination results for a class of polytopes

期刊论文
发表日期：

2024年4月8日

查看全文

Shenghua Gao,
Horst Martini,
Senlin Wu,
Longzhen Zhang

摘要

In this paper, we focus on covering and illumination properties of a specific class of convex polytopes denoted by $\mathcal {P}$ $P$ . These polytopes are obtained as the convex hull of the Minkowski sum of a finite subset of $\mathbb {Z}^n$ $Z^{n}$ and $(1/2)[-1,1]^n$ $(1 / 2) {[- 1, 1]}^{n}$ . Our investigation includes the verification of Hadwiger’s covering conjecture for $\mathcal {P}$ $P$ , as well as the estimation of the covering functional for convex polytopes in $\mathcal {P}$ $P$ . Furthermore, we demonstrate that when an integer M is sufficiently large, the elements belonging to $\mathcal {P}$ $P$ that are contained in $M[-1,1]^n$ $M {[- 1, 1]}^{n}$ serve as an $\varepsilon $ $ε$ -net for the space of convex bodies in $\mathbb {R}^n$ $R^{n}$ , equipped with the Banach–Mazur metric.