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 3dimensional 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 3dimensional 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 4dimensional Euclidean space.
 Publication:

arXiv eprints
 Pub Date:
 August 2011
 arXiv:
 arXiv:1108.6014
 Bibcode:
 2011arXiv1108.6014G
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 51M25;
 52B11;
 13P15
 EPrint:
 23 pages