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Minkowski ideals and rings

Geir Agnarsson, Jim Lawrence

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.

Minkowski ideals and rings

Abstract

\emph{Minkowski rings} are certain rings of simple functions on the Euclidean space with multiplicative structure derived from Minkowski addition of convex polytopes. When the ring is (finitely) generated by a set of indicator functions of polytopes then the ring can be presented as when viewed as a -algebra, where is the ideal describing all the relations implied by identities among Minkowski sums of elements of . We discuss in detail the -dimensional case, the -dimensional box case and the affine Coxeter arrangement in where the convex sets are formed by closed half-planes with bounding lines making the regular triangular grid in . We also consider, for a given polytope , the Minkowski ring of the collection of the nonempty faces of and their multiplicative inverses. Finally we prove some general properties of identities in the Minkowski ring of ; in particular, we show that Minkowski rings behave well under Cartesian product, namely that as -algebras where and are polytopes.

Paper Structure

This paper contains 16 sections, 23 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Definitions and observations
  3. A few simple cases
  4. The principal case
  5. A one dimensional case
  6. (A) $0 = \alpha < \beta$:
  7. (B) $0 < \alpha < \beta$ and $\alpha/\beta\not\in {\hbox{$\mathbb Q$}}$:
  8. (C) $0 < \alpha < \beta$ and $\alpha/\beta\in {\hbox{$\mathbb Q$}}$:
  9. (D) $\alpha < 0 < \beta$ and $\alpha/\beta\in {\hbox{$\mathbb Q$}}$:
  10. The positive integer box case
  11. The general integer box case
  12. The affine arrangement and identities
  13. The Coxeter arrangement
  14. Minkowski identities
  15. The Cartesian product of polytopes
  16. ...and 1 more sections

Key Result

Corollary 3.4

Let $S\subseteq W = {\hbox{$\mathbb R$}}^d$ be a closed set. For the surjection $\phi : {\hbox{$\mathbb C$}}[x]^{\pm}\twoheadrightarrow M^{\pm}(S)\subseteq M^{\pm}({\mathcal{P}}(W))$$\phi(x) = S$ we have $I = \ker(\phi)$ is as follows: If $S = \emptyset$ then $I = R^{\pm}$. If $S = \{0\}$ or if $S \

Theorems & Definitions (43)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 3.1
  • Example 3.2
  • Corollary 3.4
  • Proposition 3.5
  • Corollary 3.6
  • Claim 3.7
  • Lemma 3.8
  • ...and 33 more