The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces

Abstract

A flexible polyhedron in an -dimensional space of constant curvature is a polyhedron with rigid -dimensional faces and hinges at -dimensional faces. The Bellows conjecture claims that, for , the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces was proved by Sabitov for (1996) and by the author for (2012). Counterexamples to the Bellows conjecture in open hemispheres were constructed by Alexandrov for (1997) and by the author for (2015). In this paper we prove the Bellows conjecture for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces. The proof is based on the study of the analytic continuation of the volume of a simplex in Lobachevsky space considered as a function of the hyperbolic cosines of its edge lengths. Bibliography: 37 titles.