2017/08/26 by Alexander, Matthew, Fradelizi, Matthieu, Zvavitch, Artem · 1 citation
#52A20 #52B10 #53A15 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1708.07914
For a convex body K ⊂ \mathbb Rn, let Kz = \y∈\mathbb Rn : ⟨ y-z, x-z⟩≤ 1, for all x∈ K\ be the polar body of K with respect to the center of polarity z ∈ \mathbb Rn. The goal of this paper is to study the maximum of the volume product P(K)=min_z∈ \rm int(K)|K||Kz|, among convex polytopes K⊂ \mathbb Rn 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 \mathbb Rn.