A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes

We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P1+⋯+Pr, of r convex d-polytopes P1, . . . , Pr in R, where d ≥ 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal’s approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f -vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P1 + ⋯ + Pr as a section of the Cayley polytope C of the summands; bounding the k-faces of P1 + ⋯ + Pr reduces to bounding the subset of the (k + r − 1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems – Geometrical problems and computations, G.2.1 Combinatorics