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

Alexander A. Gaifullin

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.

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

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.

Paper Structure

This paper contains 9 sections, 20 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Polyhedra and their volumes
  3. Simplicial polyhedra
  4. Generalized oriented volume
  5. Configuration spaces of polyhedra
  6. Analytic continuation of the volume of a simplex
  7. Hereditary orderings on simplicial complexes
  8. Simplicial complexes $\mathcal{K}(G,\varkappa)$
  9. Proof of the main theorem

Key Result

Theorem \oldthetheorem

Let $\mathbb{X}^n$ be either $\mathbb{S}^n$ or $\Lambda^n,$$n\ge 3$. Let $P_t$ be a simplicial flexible polyhedron in $\mathbb{X}^n$ with $m$ vertices such that all edges of $P_t$ have lengths smaller than $2^{-m^2(n+4)}$. Then the generalized oriented volume of $P_t$ is constant during the flexion.

Theorems & Definitions (42)

  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 32 more