This book is devoted to certain problems from the theory of convex bodies (see pages 3 and 7 for the definition of a convex body). The theory of convex bodies, which was developed at the end of the last century, is today a science that is rich in general methods and important special results. It is still being intensively investigated. t The number of printed scientific articles and books concerned with this subject is so great that the theory of convex bodies occupies a place as an independent mathematical discipline among the other branches of mathematical science in the tables of contents of the modern mathematical reviewing journals in which all mathematical publications are mentioned. This popularity of the theory of convex bodies is related first of all to its importance for geometry, but is also due to the significant applications which it has found in other fields of mathematics (algebra, number theory, etc.) as well as in the natural sciences (for example, mathematical crystallography). The importance of the theory of convex bodies has especially increased since the recent outstanding works of the Leningrad mathematician, A. D. Aleksandrov, who has made this theory the basis of a new development in a very important branch of modern geometry differential geometry.