Numerical characterization of nn-cube subset partitioning

A general quantitative description of vertex subsets of the n-dimensional unit cube En through their partitions (direct problem) is given and the existence and composition problems for vertex subsets with given quantitative characteristics of partitions (inverse problem) are considered. Each of these subproblems is of significant theoretical and practical importance. Finding an efficient algorithmic solution to the inverse problem remains open. A complete and simple structural description of the numerical parameters of the unit cube partitions is presented.