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Fundam. Prikl. Mat., 1996, Volume 2, Issue 4, Pages 1235–1246 (Mi fpm195)  

This article is cited in 35 scientific papers (total in 35 papers)

The volume of polyhedron as a function of its metric

I. Kh. Sabitov

M. V. Lomonosov Moscow State University

Abstract: It is proved that the volume of any polyhedron is root of some polynomial whose coefficients are not depending on the concrete form of the polyhedron in three-space under the condition that its metric is known apriori. As consequence we have a proof of the “bellows conjecture” affirming the invariance of volume of a flexible polyhedron in the process of its flexion.

Full text: PDF file (543 kB)

Bibliographic databases:
UDC: 514.113.5
Received: 01.07.1996

Citation: I. Kh. Sabitov, “The volume of polyhedron as a function of its metric”, Fundam. Prikl. Mat., 2:4 (1996), 1235–1246

Citation in format AMSBIB
\Bibitem{Sab96}
\by I.~Kh.~Sabitov
\paper The volume of polyhedron as a function of its metric
\jour Fundam. Prikl. Mat.
\yr 1996
\vol 2
\issue 4
\pages 1235--1246
\mathnet{http://mi.mathnet.ru/fpm195}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1785783}
\zmath{https://zbmath.org/?q=an:0904.52002}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. Kh. Sabitov, “A generalized Heron–Tartaglia formula and some of its consequences”, Sb. Math., 189:10 (1998), 1533–1561  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Sabitov, IK, “The volume as a metric invariant of polyhedra”, Discrete & Computational Geometry, 20:4 (1998), 405  crossref  mathscinet  zmath  isi
    3. A. V. Astrelin, I. Kh. Sabitov, “A canonical polynomial for the volume of a polyhedron”, Russian Math. Surveys, 54:2 (1999), 430–431  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. Alexandrov, V, “Implicit function theorem for systems of polynomial equations with vanishing Jacobian and its application to flexible polyhedra and frameworks”, Monatshefte fur Mathematik, 132:4 (2001), 269  crossref  mathscinet  zmath  isi
    5. I. Kh. Sabitov, “Algorithmic solution of the problem of isometric realization for two-dimensional polyhedral metrics”, Izv. Math., 66:2 (2002), 377–391  mathnet  crossref  crossref  mathscinet  zmath  elib
    6. Alexandrov, V, “Flexible polyhedra in Minkowski 3-space”, Manuscripta Mathematica, 111:3 (2003), 341  mathscinet  zmath  isi
    7. Schlenker, JM, “The bellows conjecture”, Asterisque, 2004, no. 294, 77  mathscinet  zmath  isi
    8. Sabitov, I, “Solution of polyhedra”, Bulletin of the Brazilian Mathematical Society, 35:2 (2004), 199  crossref  mathscinet  zmath  isi
    9. Souam, R, “The Schlafli formula for polyhedra and piecewise smooth hypersurfaces”, Differential Geometry and Its Applications, 20:1 (2004), 31  crossref  mathscinet  zmath  isi
    10. C. D'Andrea, M. Sombra, “The Cayley–Menger determinant is irreducible for $n\geqslant3$”, Siberian Math. J., 46:1 (2005), 71–76  mathnet  crossref  mathscinet  zmath  isi
    11. Schlenker, JM, “A rigidity criterion for non-convex polyhedra”, Discrete & Computational Geometry, 33:2 (2005), 207  crossref  mathscinet  zmath  isi
    12. N. V. Abrosimov, “K resheniyu problemy Zeidelya ob ob'emakh giperbolicheskikh tetraedrov”, Sib. elektron. matem. izv., 6 (2009), 211–218  mathnet  mathscinet  elib
    13. Kane D., Price G.N., Demaine E.D., “A Pseudopolynomial Algorithm for Alexandrov's Theorem”, Algorithms and Data Structures, Lecture Notes in Computer Science, 5664, 2009, 435–446  crossref  mathscinet  zmath  isi
    14. I. G. Maksimov, “Study of flexible algorithmically 1-parametric polyhedra and description of a set of rigid embedded polyhedra”, Siberian Math. J., 51:6 (2010), 1081–1090  mathnet  crossref  mathscinet  isi
    15. Abrosimov N.V., “Seidel's Problem on the Volume of a Non-Euclidean Tetrahedron”, Doklady Mathematics, 82:3 (2010), 843–846  crossref  mathscinet  zmath  isi  elib
    16. Alexandrov V., “Algebra versus analysis in the theory of flexible polyhedra”, Aequationes Mathematicae, 79:3 (2010), 229–235  crossref  mathscinet  zmath  isi  elib
    17. I. Kh. Sabitov, “Algebraic methods for solution of polyhedra”, Russian Math. Surveys, 66:3 (2011), 445–505  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. D. A. Slutskiy, “An infinitesimally nonrigid polyhedron with nonstationary volume in the Lobachevskiĭ 3-space”, Siberian Math. J., 52:1 (2011), 131–138  mathnet  crossref  mathscinet  isi
    19. Alexandrov V., Connelly R., “Flexible Suspensions with a Hexagonal Equator”, Ill. J. Math., 55:1 (2011), 127–155  mathscinet  zmath  isi
    20. Itoh J.-i., Nara Ch., “Continuous Flattening of Platonic Polyhedra”, Computational Geometry, Graphs and Applications, Lecture Notes in Computer Science, 7033, eds. Akiyama J., Bo J., Kano M., Tan X., Springer-Verlag Berlin, 2011, 108–121  crossref  mathscinet  zmath  isi
    21. Jin-ichi Itoh, Chie Nara, “Continuous Flattening of a Regular Tetrahedron with Explicit Mappings”, Model. i analiz inform. sistem, 19:6 (2012), 127–136  mathnet
    22. D. I. Sabitov, I. Kh. Sabitov, “Mnogochleny ob'ema dlya nekotorykh mnogogrannikov v prostranstvakh postoyannoi krivizny”, Model. i analiz inform. sistem, 19:6 (2012), 161–169  mathnet
    23. Gaifullin A.A., Gaifullin S.A., “Deformations of Period Lattices of Flexible Polyhedral Surfaces”, Discret. Comput. Geom., 51:3 (2014), 650–665  crossref  mathscinet  zmath  isi  elib
    24. Gaifullin A.A., “Sabitov Polynomials for Volumes of Polyhedra in Four Dimensions”, Adv. Math., 252 (2014), 586–611  crossref  mathscinet  zmath  isi  elib
    25. A. A. Gaifullin, “Flexible cross-polytopes in spaces of constant curvature”, Proc. Steklov Inst. Math., 286 (2014), 77–113  mathnet  crossref  crossref  isi  elib  elib
    26. Gaifullin A.A., “Generalization of Sabitov'S Theorem To Polyhedra of Arbitrary Dimensions”, Discret. Comput. Geom., 52:2 (2014), 195–220  crossref  mathscinet  zmath  isi
    27. A. A. Gaifullin, “Embedded flexible spherical cross-polytopes with nonconstant volumes”, Proc. Steklov Inst. Math., 288 (2015), 56–80  mathnet  crossref  crossref  isi  elib
    28. A. A. Gaifullin, “The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces”, Sb. Math., 206:11 (2015), 1564–1609  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    29. Itoh J.-i., Nara Ch., “Continuous Flattening of Truncated Tetrahedra”, J. Geom., 107:1 (2016), 61–75  crossref  mathscinet  zmath  isi
    30. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175  mathnet  crossref  elib
    31. Alexander A. Gaifullin, “The bellows conjecture for small flexible polyhedra in non-Euclidean spaces”, Mosc. Math. J., 17:2 (2017), 269–290  mathnet  crossref  mathscinet
    32. D. I. Sabitov, I. Kh. Sabitov, “Kanonicheskie mnogochleny ob'ema dlya mnogogrannikov kombinatornogo tipa geksaedra”, Sib. elektron. matem. izv., 14 (2017), 1078–1087  mathnet  crossref
    33. 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
    34. Romakina L., “To the Volumes Theory of a Hyperbolic Space of Positive Curvature”, J. Geom. Graph., 22:1 (2018), 67–86  mathscinet  zmath  isi
    35. D. I. Sabitov, I. Kh. Sabitov, “Mnogochleny ob'ema dlya mnogogrannikov kombinatornogo tipa $n$-grannykh prizm v sluchayakh $n=5,6,7$”, Sib. elektron. matem. izv., 16 (2019), 439–448  mathnet  crossref
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