Polytopes of Maximal Volume Product.

For 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 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. [ABSTRACT FROM AUTHOR]