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 $P$.


First, a linear relation can involve only volumes of faces of equal dimensions because scaling a polytope by $\lambda$ multiplies the volume of a $k$-face by $\lambda^k$.

For a polytope in $\mathbb{R}^d$ there are $d$ linear relations between the volumes of facets which follow from the Minkowski relation: $$ \sum_i \mathrm{vol}(F_i) v_i = 0. $$ Here $v_i$ is the outer unit normal to the $i$-th face. If the polytope is simple, then, firstly, these are the only relations for polytopes with the same combinatorics and fixed facet normals (due to the Minkowski existence theorem), and secondly, slightly changing the normals will change the linear relations without changing the combinatorics. So, for simple polytopes there are no linear relations between the volumes of their facets.

If you take faces of dimension $k < d-1$, then there are relations coming from the Minkowski relations in faces of dimensions $k+1$. Again, they are destroyed by small perturbations of the polytope if the polytope is simple.

For non-simple polytopes the question is more delicate and I do not have a definite answer. But if you apply any affine or projective transformation, then the combinatorics of your polytope does not change while the face volumes change in a non-linear way. So I suspect that there are no linear relations in the general case as well.


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