Maximal f-vectors of Minkowski sums of large numbers of polytopes

TL;DR

This paper establishes a tight bound on the maximum number of vertices in Minkowski sums of multiple polytopes in any dimension, revealing a linear relation between face counts of sums with different numbers of summands.

Contribution

It introduces an exact formula linking face counts of Minkowski sums with varying numbers of polytopes, leading to a tight bound on the maximum vertices in sums of multiple polytopes.

Findings

Maximum vertices of Minkowski sums grow as O(n^{d-1})

Linear relation between face counts of sums with different numbers of summands

Bound is tight, with some sums reaching this maximum

Abstract

It is known that in the Minkowski sum of $r$ polytopes in dimension $d$, with $r<d$, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand. However, the number of vertices for sums of more polytopes was unknown so far. In this paper, we study sums of polytopes in general orientations, and show a linear relation between the number of faces of a sum of $r$ polytopes in dimension $d$, with $r\geq d$, and the number of faces in the sums of less than $d$ of the summand polytopes. We deduce from this exact formula a tight bound on the maximum possible number of vertices of the Minkowski sum of any number of polytopes in any dimension. In particular, the linear relation implies that a sum of $r$ polytopes in dimension $d$ has a number of vertices in $O(n^{d-1})$ of the total number of vertices in the summands, even when $r\geq d$. This bound is tight, in the sense that some sums do have that many vertices.