A substantial survey of results, obtained around the discrete isoperimetry problems, is presented in Harper’s book of 2004. But a number of important results remain not reflected. These basic results, obtained in period 1980-90, and a number of new results obtained recently are included in this text. Results are presented in terms of geometry of multidimensional unit cube. And of course the considered in this paper issue concerns the vertex-isoperimetry paradigm vs. to the edge-isoperimetry version that is also a research topic in this area. The edge-isoperimetry counterpart of these topics will be surveyed in a separate publication.