\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.
Categories: math.CO math.AC