Infinitesimal volume rigidity of convex polytopes

Determine whether the k-skeleton of every convex d-polytope is infinitesimally volume rigid in R^d for every 1 <= k <= d-2.

Background

Whiteley’s theorem, cited by the authors, establishes infinitesimal rigidity of the 1-skeleton of every convex d-polytope for d >= 3 without a genericity assumption. The paper asks whether an analogous result holds for higher-dimensional face-volume constraints.

The proposed extension concerns every intermediate skeleton from the 1-skeleton through the (d-2)-skeleton, and replaces ordinary edge-length rigidity with infinitesimal volume rigidity. The question is explicitly presented in the paper’s closing section on open problems.

References

Could this result be extended to show that the $k$-skeleton of every convex $d$-polytope is infinitesimally volume rigid in $d$ for all $1\leq k\leq d-2$?

— Volume Rigidity of Simplicial Manifolds  (2503.01647 - Cruickshank et al., 3 Mar 2025) in Section 5, Closing remarks and open problems, subsection “Infinitesimal volume rigidity of convex polyhedra”