The bellows conjecture for small flexible polyhedra in non-Euclidean spaces

TL;DR

This paper proves that the bellows conjecture holds for small flexible polyhedra in spherical and Lobachevsky spaces of dimension three or higher, extending previous results and addressing known counterexamples.

Contribution

It establishes the validity of the bellows conjecture for all small flexible polyhedra in higher-dimensional spherical and Lobachevsky spaces.

Findings

Bellows conjecture holds for small flexible polyhedra in spherical spaces.

Bellows conjecture holds for small flexible polyhedra in Lobachevsky spaces.

Counterexamples exist in open hemispheres, but not for sufficiently small polyhedra.

Abstract

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 the Euclidean spaces of dimensions 3 and higher, and for bounded flexible polyhedra in the odd-dimensional Lobachevsky spaces. Counterexamples to the bellows conjecture are known in all open hemispheres of dimensions 3 and higher. The aim of this paper is to prove that, nonetheless, the bellows conjecture is true for all flexible polyhedra in either spheres or Lobachevsky spaces of dimensions greater than or equal to 3 with sufficiently small edge lengths.