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 Trudy Mat. Inst. Steklova, 2015, Volume 288, Pages 67–94 (Mi tm3598)

Embedded flexible spherical cross-polytopes with nonconstant volumes

A. A. Gaifullinabc

a Lomonosov Moscow State University, Moscow, Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia

Abstract: We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions $4$ and higher, they are the first examples of embedded flexible polyhedra. Notice that, in contrast to the spheres, in the Euclidean and Lobachevsky spaces of dimensions $4$ and higher still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are nonconstant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was constructed only in dimension $3$ (V. A. Alexandrov, 1997), and it was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type, which includes our counterexamples to the ordinary Bellows Conjecture. Simultaneously, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write down relations for the volumes of their faces of codimensions $1$ and $2$.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-1246914-01-00537 Ministry of Education and Science of the Russian Federation ÌÄ-2969.2014.1 Dynasty Foundation

DOI: https://doi.org/10.1134/S0371968515010057

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 288, 56–80

Bibliographic databases:

UDC: 514.114

Citation: A. A. Gaifullin, “Embedded flexible spherical cross-polytopes with nonconstant volumes”, Geometry, topology, and applications, Collected papers. Dedicated to Professor Nikolai Petrovich Dolbilin on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 288, MAIK Nauka/Interperiodica, Moscow, 2015, 67–94; Proc. Steklov Inst. Math., 288 (2015), 56–80

Citation in format AMSBIB
\Bibitem{Gai15} \by A.~A.~Gaifullin \paper Embedded flexible spherical cross-polytopes with nonconstant volumes \inbook Geometry, topology, and applications \bookinfo Collected papers. Dedicated to Professor Nikolai Petrovich Dolbilin on the occasion of his 70th birthday \serial Trudy Mat. Inst. Steklova \yr 2015 \vol 288 \pages 67--94 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3598} \crossref{https://doi.org/10.1134/S0371968515010057} \elib{https://elibrary.ru/item.asp?id=23302165} \transl \jour Proc. Steklov Inst. Math. \yr 2015 \vol 288 \pages 56--80 \crossref{https://doi.org/10.1134/S0081543815010058} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000353881900005} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84928718929} 

• http://mi.mathnet.ru/eng/tm3598
• https://doi.org/10.1134/S0371968515010057
• http://mi.mathnet.ru/eng/tm/v288/p67

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This publication is cited in the following articles:
1. A. A. Gaifullin, “The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces”, Sb. Math., 206:11 (2015), 1564–1609
2. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175
3. Alexander A. Gaifullin, “The bellows conjecture for small flexible polyhedra in non-Euclidean spaces”, Mosc. Math. J., 17:2 (2017), 269–290
4. Alexander A. Gaifullin, Leonid S. Ignashchenko, “Dehn invariant and scissors congruence of flexible polyhedra”, Proc. Steklov Inst. Math., 302 (2018), 130–145
5. V. Alexandrov, “A sufficient condition for a polyhedron to be rigid”, J. Geom., 110:2 (2019), UNSP 38
6. V. Alexandrov, “The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in R-D does not always remain unaltered during the flex”, J. Geom., 111:2 (2020), 32
7. 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
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