On the properties of K. Beretsky’s partition constructions

This book is devoted to a survey of non-normal partitions of hyperbolic space, in particular a survey of irregular partitions of K. Beretsky and some useful consequences of the proposed constructions. With the help of this partition (Beretsky’s), it is easy to construct examples of non-normal partitions of n-dimensional hyperbolic space (constructive proof of the existence theorem) by equal compact convex polyhedra and these partitions cannot be transformed into regular ones by transposing the partition polyhedra. In this paper we note some possible generalisations of the construction of K. Beretsky, which, in most cases, also allow to construct non-normal partitions. The peculiarities of the partitions allow one to constructively prove some general statements concerning, for example, Delaunay point systems and Delaunay partitions. The publication also discusses the question of the number of hyperfacets of a (hyperbolic) tie.