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, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1+...+P_r$, of $r$ convex $d$-polytopes $P_1,...,P_r$ in $\mathbb{R}^d$, where $d\ge{}2$ 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 $P_1+...+P_r$ as a section of the Cayley polytope $\mathcal{C}$ of the summands; bounding the $k$-faces of $P_1+...+P_r$ reduces to bounding the subset of the $(k+r-1)$-faces of $\mathcal{C}$ 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.