On the infinitesimal rigidity of polyhedra with vertices in convex position
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 HilbertEinstein 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.1981
 Bibcode:
 2007arXiv0711.1981I
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Metric Geometry
 EPrint:
 12 pages