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

We derive tight bounds for the maximum number of k -faces, 0 ≤ k ≤ d - 1 , of the Minkowski sum, P 1 + P 2 , of two d -dimensional convex polytopes P 1 and P 2 , as a function of the number of vertices of the polytopes. For even dimensions d ≥ 2 , the maximum values are attained when P 1 and P 2 are cyclic d -polytopes with disjoint vertex sets. For odd dimensions d ≥ 3 , the maximum values are attained when P 1 and P 2 are ⌊ d 2 ⌋ -neighborly d -polytopes, whose vertex sets are chosen appropriately from two distinct d -dimensional moment-like curves.