Polytopes of Maximal Volume Product

Abstract

For a convex body $K\subset {\mathbb{R}}^n$, let $K^z = \{y\in{\mathbb R}^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of $K$ with respect to the center of polarity $z\in {\mathbb{R}}^n$. The goal of this paper is to study the maximum of the volume product $\mathcal{P}(K)={\min }_{z\in \mathrm{int}(K)}|K||K^z|$, among convex polytopes $K\subset {\mathbb{R}}^n$ with a number of vertices bounded by some fixed integer $m\ge n+1$. In particular, we prove that the supremum is reached at a simplicial polytope with exactly $m$ vertices and we provide a new proof of a result of Meyer and Reisner showing that, in the plane, the regular polygon has maximal volume product among all polygons with at most $m$ vertices. Finally, we treat the case of polytopes with $n+2$ vertices in ${\mathbb{R}}^n$.