Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions
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arXiv
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| Format: | Preprint |
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2012
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| _version_ | 1866910704371499008 |
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| author | Gaifullin, Alexander A. |
| author_facet | Gaifullin, Alexander A. |
| contents | 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 $n\ge 3$. 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 $\C^n$ from their orthogonal edge lengths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1210_5408 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions Gaifullin, Alexander A. Metric Geometry Algebraic Geometry Combinatorics 51M25, 52B11, 13A18 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 $n\ge 3$. 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 $\C^n$ from their orthogonal edge lengths. |
| title | Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions |
| topic | Metric Geometry Algebraic Geometry Combinatorics 51M25, 52B11, 13A18 |
| url | https://arxiv.org/abs/1210.5408 |