Generalization of Sabitov’s Theorem to Polyhedra of Arbitrary Dimensions

Abstract

In 1996 Sabitov proved that the volume V of an arbitrary simplicial polyhedron P in the 3-dimensional Euclidean space (Formula presented.) satisfies a monic (with respect to V) polynomial relation (Formula presented.), where (Formula presented.) denotes the set of the squares of edge lengths of P. In 2011 the author proved the same assertion for polyhedra in (Formula presented.). In this paper, we prove that the same result is true in arbitrary dimension n≥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 (Formula presented.), is a continuous deformation of a polyhedron such that the combinatorial type of (Formula presented.) does not change and every 2-face of (Formula presented.) remains congruent to the corresponding face of (Formula presented.), then the volume of (Formula presented.) is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in (Formula presented.) from their orthogonal edge lengths.