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 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 R 4 . 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 Pt, t ∈ [0, 1], is a continuous deformation of a polyhedron such that the combinatorial type of Pt does not change and every 2-face of Pt remains congruent to the corresponding face of P 0 , then the volume of Pt is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in C n from their orthogonal edge lengths.

Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex

“…However, the real flowering of the theory of flexible polyhedra began in the mid-1970s when Robert Connelly constructed a sphere-homeomorphic self-intersection free flexible polyhedron [7]. Now we already know that any flexible polyhedron in Euclidean 3-space preserves the total mean curvature (see [1]), enclosed volume (there are especially many articles devoted to this issue, so we point out several of them in chronological order: [23], [24], [25], [10], [26], [27], [14], and [13]), and Dehn invariants (see [16]). The reader can find more details about the theory of flexible polyhedra in the above mentioned articles, as well as in the following review articles, which we list in chronological order: [21], [8], [29], [28], and [15].…”

A sufficient condition for a polyhedron to be rigid

“…(d) during the flex, they necessarily keep unaltered the Dehn invariants, see [21]; (e) the notion of a flexible polyhedron can be introduced in all spaces of constant curvature of dimension 3 and above (see [36], [37], [16], [17], [19], as well as in pseudo-Euclidean spaces of dimension 3 and above (see [3]); moreover, it is known that in many of these spaces flexible polyhedra do exist and possess properties similar to properties (a)-(d) (see [24], [14], [15], [18], [20]).…”

Dehn Invariant and Scissors Congruence of Flexible Polyhedra

“…• flexible polyhedra in Euclidean spaces E n (Sabitov for n = 3, see [20], [21], [22], and see [9] for another proof, and the first listed author for n ≥ 4, see [12], [13]), • bounded flexible polyhedra in odd-dimensional Lobachevsky spaces Λ 2n+1 , see [16], • flexible polyhedra with sufficiently small edge lengths in any of the spaces Λ n and S n , see [17]. Counterexamples to the Bellows Conjecture in open hemispheres S n + were constructed by Alexandrov [1] for n = 3 and by the first listed author [15] for n ≥ 4.…”