Polytopes of Maximal Volume Product

International audienceFor a convex body K ⊂ R n , let K z = {y ∈ R n : y−z, x−z ≤ 1, for all x ∈ K} be the polar body of K with respect to the center of polarity z ∈ R n. The goal of this paper is to study the maximum of the volume product P(K) = min z∈int(K) |K||K z |, among convex polytopes K ⊂ R n with a number of vertices bounded by some fixed integer m ≥ 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 R n