Polytopes of Maximal Volume Product

For a convex body K subset of R-n, let K-z = {y is an element of R-n : < y - z, x - z <= 1, for all x is an element of K} be the polar body of K with respect to the center of polarity z. Rn. The goal of this paper is to study the maximum of the volume product P(K) = min(z is an element of int(K)) vertical bar K parallel to K-z vertical bar, among convex polytopes K subset of R-n with a number of vertices bounded by some fixed integer m >= n + 1. In particular, we prove a combinatorial formula characterizing a polytope of maximal volume product and use this formula to show that the supremum is reached at a simplicial polytope with exactly m vertices. We also use this formula to provide a 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 R-n.