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

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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 Euclidean spaces ℝn, n ≥ 3, and for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces Λ2m+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 Sn or Λn, n ≥ 3, with sufficiently small edge lengths.

Original languageEnglish
Pages (from-to)269-290
Number of pages22
JournalMoscow Mathematical Journal
Volume17
Issue number2
DOIs
Publication statusPublished - 2017

Keywords

  • Analytic continuation
  • Flexible polyhedron
  • Simplicial collapse
  • The bellows conjecture

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