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

A flexible polyhedron in an $n$-dimensional space $\mathbb{X}^n$ of constant curvature is a polyhedron with rigid $(n-1)$-dimensional faces and hinges at $(n-2)$-dimensional faces. The Bellows conjecture claims that, for $n\geqslant 3$, the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces $\mathbb{E}^n$ was proved by Sabitov for $n=3$ (1996) and by the author for $n\geqslant 4$ (2012). Counterexamples to the Bellows conjecture in open hemispheres $\mathbb{S}^n_+$ were constructed by Alexandrov for $n=3$ (1997) and by the author for $n\geqslant 4$ (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.
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