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- W2950615837 abstract "In 1996 Sabitov proved that the volume of an arbitrary simplicial polyhedron P in the 3-dimensional Euclidean space $R^3$ satisfies a monic (with respect to V) polynomial relation F(V,l)=0, where l denotes the set of the squares of edge lengths of P. In 2011 the author proved the same assertion for polyhedra in $R^4$. In this paper, we prove that the same result is true in arbitrary dimension $nge 3$. 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 $P_t$, $tin [0,1]$, 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 $C^n$ from their orthogonal edge lengths." @default.
- W2950615837 created "2019-06-27" @default.
- W2950615837 creator A5089268822 @default.
- W2950615837 date "2012-10-19" @default.
- W2950615837 modified "2023-09-27" @default.
- W2950615837 title "Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions" @default.
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