Minkowski summands of cubes

TL;DR

This paper characterizes the type cone of the product of simplices, showing it is a cone over a simplex, which simplifies understanding Minkowski summands of certain polytopes.

Contribution

It provides a novel, explicit description of the type cone for the product of simplices, extending classical results on Minkowski summands.

Findings

Type cone of product of simplices is a cone over a simplex

Explicit computation of type cones is generally difficult

New insights connect rainbow point configurations to Minkowski summands

Abstract

In pioneering works of Meyer and of McMullen in the early 1970s, the set of Minkowski summands of a polytope was shown to be a polyhedral cone called the type cone. Explicit computations of type cones are in general intractable. Nevertheless, we show that the type cone of the product of simplices is the cone over a simplex. This remarkably simple result derives from insights about rainbow point configurations and the work of McMullen.