THE BELLOWS CONJECTURE FOR SMALL FLEXIBLE POLYHEDRA IN NON-EUCLIDEAN SPACES

The bellows conjecture claims that the volume of any flexible polyhedron of dimension 3 or higher is constant during the flexion. The bellows conjecture was proved for flexible polyhedra in Euclidean spaces R-n, n >= 3, and for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces A(2m+1), m >= 1. Counterexamples to the bellows conjecture are known in all open hemispheres S-n, n >= 3. The aim of this paper is to prove that, nonetheless, the bellows conjecture is true for all flexible polyhedra in either S-n or A(n), n >= 3, with sufficiently small edge lengths.