Discrete & Computational Geometry

, Volume 52, Issue 2, pp 195–220 | Cite as

Generalization of Sabitov’s Theorem to Polyhedra of Arbitrary Dimensions

Article

Abstract

In 1996 Sabitov proved that the volume \(V\) of an arbitrary simplicial polyhedron \(P\) in the \(3\)-dimensional Euclidean space \(\mathbb {R}^3\) satisfies a monic (with respect to \(V\)) polynomial relation \(F(V,\ell )=0\), where \(\ell \) denotes the set of the squares of edge lengths of \(P\). In 2011 the author proved the same assertion for polyhedra in \(\mathbb {R}^4\). In this paper, we prove that the same result is true in arbitrary dimension \(n\ge 3\). Moreover, we show that this is true not only for simplicial polyhedra, but for all polyhedra with triangular \(2\)-faces. As a corollary, we obtain the proof in arbitrary dimension of the well-known Bellows Conjecture posed by Connelly in 1978. This conjecture claims that the volume of any flexible polyhedron is constant. Moreover, we obtain the following stronger result. If \(P_t, t\in [0,1]\), is a continuous deformation of a polyhedron such that the combinatorial type of \(P_t\) does not change and every \(2\)-face of \(P_t\) remains congruent to the corresponding face of \(P_0\), then the volume of \(P_t\) is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in \(\mathbb {C}^n\) from their orthogonal edge lengths.

Keywords

Flexible polyhedron Sabitov polynomial Bellows conjecture  A place of field Simplicial collapse 

Notes

Acknowledgments

The work was partially supported by the Russian Foundation for Basic Research (projects 12-01-31444 and 11-01-00694), by a grant of the President of the Russian Federation (projects MD-4458.2012.1 and MD-2969.2014.1), by a grant of the Government of the Russian Federation (project 2010-220-01-077), by a programme of the Branch of Mathematical Sciences of the Russian Academy of Sciences, and by a grant from Dmitry Zimin’s “Dynasty” foundation. The author is grateful to I. I. Bogdanov, S. O. Gorchinsky, I. Pak, and I. Kh. Sabitov for useful discussions.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscow Russia
  2. 2.Moscow State UniversityMoscowRussia
  3. 3.Kharkevich Institute for Information Transmission ProblemsMoscowRussia
  4. 4.Yaroslavl State UniversityYaroslavl’Russia

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