  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.
