Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums

TL;DR

This paper establishes tight upper bounds for the faces of Minkowski sums of polytopes, generalizing classical theorems, using relative Stanley-Reisner theory to connect combinatorial and topological properties.

Contribution

It introduces a tight upper bound theorem for Minkowski sums' faces and mixed faces, utilizing relative Stanley-Reisner theory for the first time in this context.

Findings

Tight upper bounds for face counts of Minkowski sums

Characterization of equality cases in bounds

Applications to isoperimetric inequalities

Abstract

In this paper we settle long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed faces of Minkowski sums. This has a wide range of applications and generalizes the classical the Upper Bound Theorems of McMullen and Stanley. Our main tool is relative Stanley--Reisner theory, a powerful generalization of the algebraic theory of simplicial complexes inaugurated by Hochster, Reisner, and Stanley. A key feature of our theory is the ability to accomodate topological as well as combinatorial restrictions. We illustrate this by providing several simplicial isoperimetric and reverse isoperimetric inequalities.