  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.
