Dedicated to the modern problems of geometry . Issues close to the Helli's khic theorem and related to the properties of the translatable family (t.) are considered. e. (with parallel transfers) convex bodies. The authors give a complete overview (from moss.) Proof) of results in this field of combinatorial geometry, starting from the classical work of the famous Hungarian mathematician B. Secefalvi Nadia (1954). ) and until the most recent works. Despite the monographic nature of the publication, some attention is paid to other problems of combinatorial geometry (in particular, the problem of illumination of convex bodies, the problem of homothetic coating, etc.). ). To understand the material requires only the possession of elements of multidimensional geometry and the basic concepts of the theory of convex bodies in finite-dimensional space. For researchers, graduate students and students interested in bulge theory, combinatorial geometry and functional analysis.