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 ℝd, 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 [2]. 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 and shellings, and generalizes the methodology used in [15] and [14] 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.