The Maximum Number of Faces of the Minkowski Sum of Two Convex Polytopes

Abstract

We derive tight bounds for the maximum number of k-faces, $$0\le k\le d-1$$0≤k≤d-1, of the Minkowski sum, $$P_1+P_2$$P1+P2, of two d-dimensional convex polytopes $$P_1$$P1 and $$P_2$$P2, as a function of the number of vertices of the polytopes. For even dimensions $$d\ge 2$$d?2, the maximum values are attained when $$P_1$$P1 and $$P_2$$P2 are cyclic d-polytopes with disjoint vertex sets. For odd dimensions $$d\ge 3$$d?3, the maximum values are attained when $$P_1$$P1 and $$P_2$$P2 are $$\lfloor \frac{d}{2}\rfloor $$?d2?-neighborly d-polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.