A Rigidity Criterion for Non-Convex Polyhedra

Let P be a (non necessarily convex) embedded polyhedron in R, with its vertices on an ellipsoid. Suppose that the interior of P can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C1, · · · , Cn with disjoint interiors, whose vertices are the vertices of P . Suppose that there exists an ellipsoid which contains no vertex of P but intersects all the edges of the Ci. Then P is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra.