Tight lower bounds on the number of faces of the Minkowski sum of convex polytopes via the Cayley trick

Consider a set of r convex d-polytopes P1, P2, . . . , Pr, where d ≥ 3 and r ≥ 2, and let ni be the number of vertices of Pi, 1 ≤ i ≤ r. It has been shown by Fukuda and Weibel [4] that the number of k-faces of the Minkowski sum, P1 + P2 + · · · + Pr, is bounded from above by Φk+r(n1, n2, . . . , nr), where Φl(n1, n2, . . . , nr) = ∑ 1≤si≤ni s1+...+sr=l ∏ r i=1 ( ni si ) , l ≥ r. Fukuda and Weibel [4] have also shown that the upper bound mentioned above is tight for d ≥ 4, 2 ≤ r ≤ ⌊ 2 ⌋, and for all 0 ≤ k ≤ ⌊ 2 ⌋ − r. In this paper we construct a set of r neighborly d-polytopes P1, P2, . . . , Pr, where d ≥ 3 and 2 ≤ r ≤ d − 1, for which the upper bound of Fukuda and Weibel is attained for all 0 ≤ k ≤ ⌊ 2 ⌋ − r. A direct consequence of our result is a tight asymptotic bound on the complexity of the Minkowski sum P1 + P2 + · · · + Pr, for any fixed dimension d and any 2 ≤ r ≤ d− 1, when the number of vertices of the polytopes is (asymptotically) the same. Our approach is based on what is known as the Cayley trick for Minkowski sums: the Minkowski sum, P1+P2+ . . .+Pr, is the intersection of the Cayley polytope P , in R ×R, of the d-polytopes P1, P2, . . . , Pr, with an appropriately defined d-flatW of R ×R. To prove our bounds, we construct the d-polytopes P1, P2, . . . , Pr, where d ≥ 3 and 2 ≤ r ≤ d−1, in such a way so that the number of (k−1)-faces of P , that intersect the d-flatW , is equal to Φk(n1, n2, . . . , nr), for all r ≤ k ≤ ⌊ 2 ⌋. The tightness of our bounds then follows from the Cayley trick: the (k + r − 1)-faces of the intersection of P with W are in one-to-one correspondence with the k-faces of P1 + P2 + · · ·+ Pr, which implies that fk(P1 + P2 + · · ·+ Pr) = Φk+r(n1, n2, . . . , nr), for all 0 ≤ k ≤ ⌊ 2 ⌋ − r.