## Linear relations between volume of a polytope and its faces

Let $$P$$ be a polytope. Is anything known about the set of linear relations that hold between the volumes of the (not-necessarily proper) faces of $$P$$ as $$P$$ “varies slightly”? By varies slightly I mean without changing the face lattice—so, it makes sense for a linear functional to vanish at each vector $$(vol(F))_{F \text{ a face of }P’}$$ for $$P’$$ “close” to $$P$$.