There are many elegant results in the theory of convex bodies that may be fully understood by high school students, and at the same time be of interest to expert mathematicians. The aim of this book is to present some of these results. We shall discuss combinatorial problems of the theory of convex bodies, mainly connected with the partition of a set into smaller parts.

The theorems and problems in the book are fairly recent: the oldest of them is just over thirty years old, and many of the theorems are still in their infancy. They were published in professional mathematical journals during the last five years.

We consider the main part of the book to be suitable for high school students interested in mathematics. The material indicated as complicated may be skipped by them. The most straightforward sections concern plane sets: §§1–3, 7–10, 12–14. The remaining sections relate to spatial (and even n-dimensional) sets. For the keen and well-prepared reader, at the end of the book will be found notes, as well as a list of journals, papers and books. References to the notes are given in round brackets (). and references to the bibliography in square brackets []. In several places, especially in the notes, the discussion is at the level of scientific papers. We did not consider it inappropriate to include such material in a non-specialized book.

This book originally appeared in Russian almost twenty years ago; nevertheless it is as fresh now as then. No better exposition of the main results has since appeared, and the problems stated at the end of the book still remain unsolved.

I would like to mention two books which appeared after this volume and which are closely related to this material. The first is “The Decomposition of Figures into Smaller Parts” by the same authors, which appeared in English translation in the University of Chicago Press in 1980, and also the book of V.G. Boltyansky and P.S. Soltan “Combinatorial Geometry of Different Classes of Convex Sets” Stiintsa, Kishinev, 1978 (in Russian). The first book is a popular book devoted only to combinatorial problems of the plane, and the second book is on the level of mathematical research monographs.

Finally, I would like to thank Cambridge University Press and Dr. David Tranah for their interest and cooperation.