The bellows conjecture claims that the volume of any flexible polyhedron ofdimension 3 or higher is constant during the flexion. The bellows conjecturewas proved for flexible polyhedra in the Euclidean spaces of dimensions 3 andhigher, and for bounded flexible polyhedra in the odd-dimensional Lobachevskyspaces. Counterexamples to the bellows conjecture are known in all openhemispheres of dimensions 3 and higher. The aim of this paper is to prove that,nonetheless, the bellows conjecture is true for all flexible polyhedra ineither spheres or Lobachevsky spaces of dimensions greater than or equal to 3with sufficiently small edge lengths.
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