Sabitov polynomials for volumes of polyhedra in four dimensions

TL;DR

This paper extends Sabitov's polynomial volume invariance results from 3D to 4D Euclidean space, showing that polyhedron volumes are roots of specific polynomials depending only on combinatorics and edge lengths.

Contribution

It proves that Sabitov's polynomial volume invariance results hold for 4-dimensional Euclidean polyhedra, generalizing previous 3D findings.

Findings

Volume of 4D polyhedra is a root of a polynomial depending on edge lengths.

Volumes are finitely many for fixed combinatorial type and edge lengths.

Flexible 4D polyhedra have constant volume.

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.