1 research outputs found
Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions
In 1996 Sabitov proved that the volume of an arbitrary simplicial polyhedron
P in the 3-dimensional Euclidean space 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 . In this paper, we prove that the same result is true in arbitrary
dimension . 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 , , is a continuous deformation of a polyhedron
such that the combinatorial type of does not change and every 2-face of
remains congruent to the corresponding face of , then the volume of
is constant. We also obtain non-trivial estimates for the oriented
volumes of complex simplicial polyhedra in \C^n from their orthogonal edge
lengths.Comment: 21 pages, 1 figur