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$ .