On the infinitesimal rigidity of polyhedra with vertices in convex position

Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability. The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite.

💡 Research Summary

The paper addresses a long‑standing conjecture concerning the infinitesimal rigidity of three‑dimensional polyhedra whose vertices lie on the boundary of a strictly convex domain (the “weakly convex” condition) and which admit a triangulation without introducing new vertices (the “decomposable” condition). While Cauchy’s classical theorem guarantees rigidity for strictly convex polyhedra, the conjecture asks whether the weaker hypothesis together with decomposability already forces infinitesimal rigidity. The authors prove the conjecture under an additional, natural geometric assumption called codecomposability.

Codecomposability means that the interior triangulation of the polyhedron and its outer convex hull fit together in a complementary way: each interior edge belongs to exactly two tetrahedra, and interior faces never intersect the exterior faces. This condition is satisfied by a wide class of polyhedral structures that appear in architectural and engineering applications, where an interior “framework” is inserted into a convex shell.

The central technical tool is the Hilbert–Einstein functional (S), defined on the space of interior edge lengths ({\ell_i}) by
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