All polytopes here are assumed to be convex lattice polytopes.
Given a polytope $P$, set $$v(P):= (\operatorname{vol}(F))_{F\text{ a face of }P},$$ where the volume of a $d$-dimensional polytope $P\subseteq \Bbb R^n$ is defined to be its ($d$-dimensional) Euclidean volume divided by the determinant of the lattice induced by $P$.
Let $\mathcal P$ be a polytopal cell complex, $m$ the total number of faces. For each polytope $P$ isomorphic to $\mathcal P$ as a polytopal cell complex, fix an identification of the faces of $P$ with those of $\mathcal P$, so that all the vectors $v(P)$ for such $P$ live in the same vector space $\Bbb R^m$ (after choosing a linear order for the faces). Define $v(\mathcal P)$ to be the set of all such $v(P)$, and let $$v(\mathcal P)^\vee:= \{x\in \Bbb R^m \mid x\cdot v \geq 0 \text{ for all }v\in v(\mathcal P)\}.$$
Questions:
Is $v(\mathcal P)^\vee$ strongly convex? If not, what is its lineality space?
Is $v(\mathcal P)^\vee$ polyhedral?
