Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions

Abstract

In 1996 Sabitov proved that the volume 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,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 ${\mathbb{R}}^4$. 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 $\C^n$ from their orthogonal edge lengths.

21 pages, 1 figure

References in corpus (2)

• Sabitov polynomials for volumes of polyhedra in four dimensions
• Flexible cross-polytopes in spaces of constant curvature

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• Deformations of period lattices of flexible polyhedral surfaces
• Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities
• The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in ${\mathbb{R}}^d$ does not always remain unaltered during the flex
• How many times can the volume of a convex polyhedron be increased by isometric deformations?
• The volume of an infinitesimally flexible polyhedron is a multiple root of its volume polynomial