Discrete & Computational Geometry
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 Discrete Comput. Geom., 2014, Volume 52, Issue 2, Pages 195–220 (Mi dcg3)

Generalization of Sabitov's theorem to polyhedra of arbitrary dimensions

A. A. Gaifullinabcd

a Moscow State University, Moscow, Russia
b Yaroslavl State University, Yaroslavl’, Russia
c Steklov Mathematical Institute, Gubkina str. 8, Moscow 119991, Russia
d Kharkevich Institute for Information Transmission Problems, Moscow, Russia

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 $n\geqslant3$. 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.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-01-3144411-01-00694 Ministry of Education and Science of the Russian Federation MD-4458.2012.1MD-2969.2014.12010-220-01-077 Russian Academy of Sciences - Federal Agency for Scientific Organizations Dynasty Foundation 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.

DOI: https://doi.org/10.1007/s00454-014-9609-2

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Accepted:19.06.2014
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