Minkowski ideals and rings

TL;DR

This paper explores Minkowski rings, algebraic structures derived from convex polytopes and Minkowski addition, analyzing their properties, specific cases, and behavior under Cartesian products.

Contribution

It introduces the concept of Minkowski rings, studies their structure in various cases, and proves their behavior under Cartesian products, advancing the algebraic understanding of convex geometry.

Findings

Minkowski rings can be presented as polynomial quotients with relations from Minkowski sums.

Detailed analysis of 1D, 2D box, and Coxeter arrangement cases.

Minkowski rings of product polytopes decompose as tensor products.

Abstract

\emph{Minkowski rings} are certain rings of simple functions on the Euclidean space $W = {\mathbb{R}}^d$ with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set ${\cal{P}}$ of indicator functions of $n$ polytopes then the ring can be presented as ${\mathbb{C}}[x_1,\ldots,x_n]/I$ when viewed as a ${\mathbb{C}}$-algebra, where $I$ is the ideal describing all the relations implied by identities among Minkowski sums of elements of ${\cal{P}}$. We discuss in detail the $1$-dimensional case, the $d$-dimensional box case and the affine Coxeter arrangement in ${\mathbb{R}}^2$ where the convex sets are formed by closed half-planes with bounding lines making the regular triangular grid in ${\mathbb{R}}^2$. We also consider, for a given polytope $P$, the Minkowski ring $M^\pm_F(P)$ of the collection ${\cal{F}}(P)$ of the nonempty faces of $P$ and their multiplicative inverses. Finally we prove some general properties of identities in the Minkowski ring of ${\cal{F}}(P)$; in particular, we show that Minkowski rings behave well under Cartesian product, namely that $M^\pm_F(P\times Q) \cong M^{\pm}_F(P)\otimes M^{\pm}_F(Q)$ as ${\mathbb{C}}$-algebras where $P$ and $Q$ are polytopes.