The Grothendieck group of polytopes and norms

TL;DR

This paper explores the Grothendieck group formed from polytopes with integral vertices under Minkowski sum, demonstrating that all symmetric polytopes are norms within this group for any dimension.

Contribution

It establishes that every symmetric polytope can be regarded as a norm in the Grothendieck group of integral polytopes across all dimensions.

Findings

Every symmetric polytope is a norm in the Grothendieck group.

The structure applies to polytopes in any dimension.

Provides a new perspective on polytopes and norms.

Abstract

Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.