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 \({{\mathbb {C}}}[x_1,\ldots ,x_n]/I\) 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 \({\mathcal {P}}\) . 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 \(M^\pm _F(P)\) 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 \(M^\pm _F(P\times Q) \cong M^{\pm }_F(P)\otimes M^{\pm }_F(Q)\) as \({{\mathbb {C}}}\) -algebras where P and Q are polytopes.
