On the infinitesimal rigidity of polyhedra with vertices in convex position

TL;DR

This paper proves a conjecture that weakly convex, decomposable polyhedra with vertices on a convex boundary are infinitesimally rigid, using properties of the Hilbert-Einstein function's Hessian.

Contribution

It establishes the infinitesimal rigidity of a class of polyhedra under a new weak assumption of codecomposability, advancing understanding in geometric rigidity theory.

Findings

The Hessian of the Hilbert-Einstein function is negative definite when there are no interior vertices.

The conjecture holds for weakly convex, decomposable polyhedra with an additional codecomposability condition.

The proof introduces new insights into the signature of the Hessian related to polyhedral deformations.

Abstract

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.