New covering and illumination results for a class of polytopes

Abstract

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