# The ring generated by a convex polytope and its faces

Let $$V=\Bbb R^n$$. Morelli defined the (commutative unital) ring $$L(V)$$ to be the additive group generated by the indicator functions of convex polytopes in $$V$$ with multiplication induced by Minkowski sum.

Let $$P$$ be a convex polytope in $$V$$, and consider the subring $$A(P)$$ of $$L(V)$$ generated by the faces of $$P$$ (including $$P$$ itself).

Question: Has $$A(P)$$ been studied at all?

• Beside (and probably before) Morelli the "big" algebraic structure was studied by Peter McMullen (algebra of polytopes) and Pukhlikov - Khovanskii (virtual polytopes). – Ivan Izmestiev Nov 20 '18 at 18:32