Generalization of Sabitov's theorem to polyhedra of arbitrary dimensions

In 1996 Sabitov proved that the volume $V$ of an arbitrary simplicial polyhedron $P$ in the $3$-dimensional Euclidean space $\mathbb{R}^3$ satisfies a monic (with respect to $V$) polynomial relation $F(V,\ell ) = 0$, where $\ell$ 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 $n\geqslant3$. 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 $\mathbb{C}^n$ from their orthogonal edge lengths.