A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes

TL;DR

This paper presents a purely geometric method to derive tight upper bounds on the number of faces of Minkowski sums of convex polytopes, generalizing previous approaches and confirming recent results.

Contribution

It introduces a geometric approach using $f$- and $h$-vector calculus and shellings to establish tight face count bounds for Minkowski sums, extending prior combinatorial algebra methods.

Findings

Derived explicit formulas for maximum face counts of Minkowski sums.

Confirmed recent bounds by Adiprasito and Sanyal through a geometric approach.

Provided a construction demonstrating the bounds' tightness.

Abstract

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.