Minkowski ideals and rings

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 $\mathcal{P}$ of indicator functions of $n$ polytopes then the ring can be presented as $\mathbb{C}[x_1,\dots ,x_n]/I$ 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 $\mathcal{P}$. 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 $M_F^{\pm }(P)$ of the collection $\mathcal{F}(P)$ of the nonempty faces of $P$ and their multiplicative inverses. Finally we prove some general properties of identities in the Minkowski ring of $\mathcal{F}(P)$; in particular, we show that Minkowski rings behave well under Cartesian product, namely that $M_F^{\pm }(P\times Q)\cong M_F^{\pm }(P)\otimes M_F^{\pm }(Q)$ as $\mathbb{C}$-algebras where $P$ and $Q$ are polytopes.

39 pages, comments and related references welcomed