  A flexible polyhedron in an n-dimensional space of constant curvature,
namely, in the Euclidean space, or in the Lobachevsky space, or in the sphere,
is a polyhedron with rigid (n-1)-dimensional faces and hinges at
(n-2)-dimensional faces. The Bellows conjecture claims that, for n greater than
or equal to 3, the volume of any flexible polyhedron is constant during the
flexion. The Bellows conjecture in Euclidean spaces was proved by Sabitov in
dimension 3 (1996) and by the author in dimensions 4 and higher (2012).
Counterexamples to the Bellows conjecture in open hemispheres were constructed
by Alexandrov in dimension 3 (1997) and by the author in dimensions 4 and
higher (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 the
Lobachevsky space considered as a function of the hyperbolic cosines of its
edge lengths.
