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Mosc. Math. J., 2017, Volume 17, Number 2, Pages 269–290 (Mi mmj632)  

This article is cited in 3 scientific papers (total in 3 papers)

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

Alexander A. Gaifullin

Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, Moscow, 119991, Russia

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 $\mathbb R^n$, $n\ge3$, and for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces $\Lambda^{2m+1}$, $m\ge1$. Counterexamples to the bellows conjecture are known in all open hemispheres $\mathbb S^n_+$, $ n\ge3$. The aim of this paper is to prove that, nonetheless, the bellows conjecture is true for all flexible polyhedra in either $\mathbb S^n$ or $\Lambda^n$, $n\ge3$, with sufficiently small edge lengths.

Key words and phrases: flexible polyhedron, the bellows conjecture, simplicial collapse, analytic continuation.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.17323/1609-4514-2017-17-2-269-290

Full text: http://www.mathjournals.org/.../2017-017-002-005.html
References: PDF file   HTML file

Bibliographic databases:

ArXiv: 1605.04568
MSC: Primary 52C25; Secondary 51M25, 05E45, 32D99
Received: May 15, 2016; in revised form December 26, 2016
Language:

Citation: Alexander A. Gaifullin, “The bellows conjecture for small flexible polyhedra in non-Euclidean spaces”, Mosc. Math. J., 17:2 (2017), 269–290

Citation in format AMSBIB
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\paper The bellows conjecture for small flexible polyhedra in non-Euclidean spaces
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\pages 269--290
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Alexander A. Gaifullin, Leonid S. Ignashchenko, “Dehn invariant and scissors congruence of flexible polyhedra”, Proc. Steklov Inst. Math., 302 (2018), 130–145  mathnet  crossref  crossref  mathscinet  isi  elib
    2. V. Alexandrov, “A sufficient condition for a polyhedron to be rigid”, J. Geom., 110:2 (2019), UNSP 38  crossref  mathscinet  isi  scopus
    3. V. A. Krasnov, “Ob'emy mnogogrannikov v neevklidovykh prostranstvakh postoyannoi krivizny”, Algebra, geometriya i topologiya, SMFN, 66, no. 4, Rossiiskii universitet druzhby narodov, M., 2020, 558–679  mathnet  crossref
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