Polytopes of Maximal Volume Product

TL;DR

This paper investigates the maximum volume product of convex polytopes with a bounded number of vertices, establishing that simplicial polytopes attain this maximum and providing new proofs for known planar cases.

Contribution

It proves that the maximum volume product among such polytopes is achieved by simplicial polytopes with a fixed number of vertices, and extends results to polytopes with n+2 vertices.

Findings

Maximum volume product is attained at simplicial polytopes.

Regular polygons have maximal volume product among polygons with limited vertices.

Results are extended to polytopes with n+2 vertices in R^n.

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 $\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K||K^z|$, 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 ${\mathbb R}^n$.