## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

On the recent achievements in the metric theory of polyhedra

On the recent achievements in the metric theory of polyhedra

Idjad Kh. Sabitov, Moscow State University

It is planed to give a survey of works concerning:

1) A solution of the "bellows conjecture" affirming the invariance of the volume of a flexible polyhedron in the process of its flexion. This solution is based on the existence of a polynomial equation for the volume of a polyhedron which can be considered as a vast generalization of Heron's formule for the area of a triangle. The newest results are: a) an algorithm for finding of the minimal degree polynomial equation for the volume of polyhedra homeomorphic to a sphere; b) a proof of the existence of a polynomial equation for the volume of polyhedra in four-dimensional space.

2) "The solution of polyhedra". This term is an analogue of the known one "the solution of triangles" and it means that we can indicate algorithms admitting to find for a polyhedron (in a general position) with a given combinatorial structure and metric not only its volume but also its diagonals and to test its flexibility/rigidity. So the metric theory of polyhedra is becoming a finite-calculated theory (one can say in the same sense as the chess is a finite-calculated play).

3) Others results relating isometric deformations and volumes of polyhedra under different conditions imposed on the deformations.

4) The problem of a correct definition of high-order rigidity and extension of an infinitesimal deformation up to a flexion.

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