Possible volumes of lattice polytopes

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:

1. Is $$v(\mathcal P)^\vee$$ strongly convex? If not, what is its lineality space?

2. Is $$v(\mathcal P)^\vee$$ polyhedral?