
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. Gaifullin^{abcd} ^{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 wellknown 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 nontrivial estimates for the oriented volumes of complex simplicial polyhedra in $\mathbb{C}^n$ from their orthogonal edge lengths.
DOI:
https://doi.org/10.1007/s0045401496092
Bibliographic databases:
Received: 23.12.2012 Accepted:19.06.2014
Language:
Linking options:
http://mi.mathnet.ru/eng/dcg3
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  79 
