Аннотация:
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.
The work was partially supported by the Russian Foundation for Basic Research (projects 10-01-92102 and 11-01-00694), by a grant of the Government of the Russian Federation (project 11.G34.31.0005), by a grant from Dmitri Ziminʼs “Dynasty” foundation and by a programme of the Branch of Mathematical Sciences of the Russian Academy of Sciences.
Поступила в редакцию: 22.10.2011 Принята в печать: 18.11.2013
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