Sabitov polynomials for volumes of polyhedra in four dimensions

Abstract

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.

23 pages; misprints corrected, Lemma 6.1 slightly rewritten, title changed

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• The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces
• Dehn invariant of flexible polyhedra
• Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex
• A sufficient condition for a polyhedron to be rigid
• Deformations of period lattices of flexible polyhedral surfaces
• The set of flexible nondegenerate polyhedra of a prescribed combinatorial structure is not always algebraic
• The spectrum of the Laplacian in a domain bounded by a flexible polyhedron in ${\mathbb{R}}^d$ does not always remain unaltered during the flex