Hyperplane sections of polyhedra, toroidal manifolds, and discrete groups in Lobachevskii space

A bounded polyhedron is called simple, if it is the intersection of half-spaces in general position. In this paper we estimate the number and the proportion of k-dimensional faces of a simple n-dimensional polyhedron, which can intersect a hyperplane not passing through a vertex of it. Here is an example of a result: a generic hyperplane cannot intersect more than P/3 + 2 edges of a simple three-dimensional polyhedron with P edges. This estimate is sharp, i.e., there exist polyhedra (with arbitrarily large number of edges) and their sections, for which the bound is achieved.