Infinitesimal rigidity of polyhedra with vertices in convex position

Ivan Izmestiev and Jean-Marc Schlenker

Vol. 248 (2010), No. 1, 171–190

DOI: 10.2140/pjm.2010.248.171

Abstract

Let P ⊂ ℝ³ be a polyhedron. It was conjectured that if P is weakly convex (that is, its vertices lie on the boundary of a strictly convex domain) and decomposable (that is, 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 about 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.

Keywords

rigidity, polyhedra, nonconvex, Hilbert–Einstein functional

Mathematical Subject Classification 2000

Primary: 52B10, 52C25

Milestones

Received: 22 August 2009

Accepted: 28 November 2009

Published: 1 October 2010