Generalization of Sabitov’s Theorem to Polyhedra of Arbitrary Dimensions

Abstract

In 1996 Sabitov proved that the volume $$V$$ V of an arbitrary simplicial polyhedron $$P$$ P in the $$3$$ 3 -dimensional Euclidean space $$\mathbb {R}^3$$ R 3 satisfies a monic (with respect to $$V$$ V ) polynomial relation $$F(V,\ell )=0$$ F ( V , l ) = 0 , where $$\ell $$ l denotes the set of the squares of edge lengths of $$P$$ P . In 2011 the author proved the same assertion for polyhedra in $$\mathbb {R}^4$$ R 4 . In this paper, we prove that the same result is true in arbitrary dimension $$n\ge 3$$ n ? 3 . Moreover, we show that this is true not only for simplicial polyhedra, but for all polyhedra with triangular $$2$$ 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 $$P_t, t\in [0,1]$$ P t , t ? [ 0 , 1 ] , is a continuous deformation of a polyhedron such that the combinatorial type of $$P_t$$ P t does not change and every $$2$$ 2 -face of $$P_t$$ P t remains congruent to the corresponding face of $$P_0$$ P 0 , then the volume of $$P_t$$ P t is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in $$\mathbb {C}^n$$ C n from their orthogonal edge lengths.