This is related to another question of mine
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.
Question: Are there any positive-dimensional polytopes $P$ satisfy the following condition (*)? If there are, is there a “good” (take that to mean what you will) characterization of them?
(*) Viewed as elements of $L(V)$, the set of faces of $P$ (including $P$ itself) is algebraically independent (over $\Bbb Z$).
Example: (*) is always false for zonotopes essentially by definition.