On the infinitesimal rigidity of weakly convex polyhedra

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by "denting" at most two edges at a common vertex, and suspensions with a natural subdivision. 1 A question on the rigidity of polyhedra A question. The rigidity of Euclidean polyhedra has a long and interesting history. Legendre [LegII] and Cauchy [Cau13] proved that convex polyhedra are rigid: if there is a continuous map between the surfaces of two convex polyhedra that is a congruence when restricted to each face, then the map is a congruence between the polyhedra (see [Sab04]). However the rigidity of non-convex polyhedra remained an open question until the first example of flexible (non-convex) polyhedra were discovered [Con77]. We say that a polyhedral su...

• Type: Journal
• Year: 2010
• Where: EJC
• Authors: Robert Connelly, Jean-Marc Schlenker
• Keywords: Convex Polyhedra, Convex Polyhedron, EJC 2007, EJC 2010, Polyhedra