Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynom...

Full description

Saved in:
Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2014-02, Vol.252, p.586-611
Main Author: Gaifullin, Alexander A.
Format: Article
Language:English
Subjects:
Cayley–Menger determinant
Flexible polyhedron
Place
Resultant
Simplicial complex
Volume
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic 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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2013.11.005