Open Collections

Description

For a convex body $K \subset \R^n$, let $$K^z = \{y\in \R^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$$ be the polar body of $K$ with respect to the center of polarity $z \in \R^n$. In this talk we would like to discuss 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 will show 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.