High dimensional flexible polyhedra
by
Victor Alexandrov
Sobolev Institute of Mathematics

A polyhedron is said to be flexible if one can change its spacial shape by means of a continuous deformation of its dihedral angles which preserves intrinsic metric of the polyhedron.

According to Cauchy theorem, there is no flexible convex polyhedron in the Euclidean 3-space. Since 1980, Steffen example is known of a flexible homeomorphic to the sphere polyhedron with 9 vertices and without self intersections. The most striking result in this field was obtained by I. Sabitov who has proven in 1996 that the volume of each flexible polyhedron remains constant during the flex.

In the present talk we are going to show that flexible polyhedra (with self intersections) do really exist in n-dimensional Euclidean, Lobachevsky and spherical spaces for each n >= 3.

This research is partially supported by the Russian Foundation for Basic Research grant no. 98-01-00688 and by the INTAS-RFBR grant IR-97-1778.

Date received: June 15, 1999