A rigidity criterion for non-convex polyhedra

2003/01/30 by Jean-Marc Schlenker
Mathematics · #math.DG #math.MG

paper · pdf

published as Discrete and Computational Geometry, 33 (2005):2, 207-221. · 11 pages, 1 image. Revised versions will be posted on http://picard.ups-tlse.fr/~schlenker v2: one statement corrected, ref. added

arxiv created 2003/01/30 · arxiv updated 2009/11/30

Abstract

Let P be a (non necessarily convex) embedded polyhedron in \R³, with its vertices on an ellipsoid. Suppose that the interior of P can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C₁, ..., C_n with disjoint interiors, whose vertices are the vertices of P. Suppose that there exists an ellipsoid which contains no vertex of P but intersects all the edges of the C_i. Then P is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra.

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