This almost 600 page monograph introduces the reader to the present state of the theory of convex geometry in its many facets. There is a carefully written account of the development of different notions and their roots; the list of references contains more than a thousand items. The book consists of four parts: Convex functions, Convex bodies, Convex polytopes and Geometry of numbers and aspects of discrete geometry. The first part consists of two chapters devoted to convex functions of one or several variables. Even in this part (which is covered in many books and which serves as preparatory material for the book), the author shows his experience and presents an excellent exposition. All proofs are carefully chosen and presented with all details. The second part of the book deals with the theory of convex bodies. It consists of 11 chapters and it covers many topics, including mixed volumes, the Brunn-Minkowski inequality, isoperimetric inequalities, symmetrization and approximation of convex bodies and the space of convex bodies. In the third part (comprising seven chapters), the author studies combinatorial properties of convex polytopes, Hilbert’s third problem, isoperimetric problems for polytopes and lattice polytopes. The last chapter is devoted to linear optimization. The most voluminous fourth part (14 chapters) includes a chapter on the Minkowski-Hlawka theorem and a chapter on problems from the geometry of numbers; a great deal of this part is devoted to packing, covering and tiling problems.
The book will be appreciated by specialists in the field but not only by them. The author relates the material to other areas of mathematics and shows numerous applications of studied theories, which makes the reading of the book useful for a much broader section of the mathematical community. The book can be used as a base for some special courses at universities and for further preparation of PhD students. It is a book that should be available in any mathematically oriented library. There are many reasons why a reader should appreciate the book. I would highlight three of them. The style of exposition reflects the author’s experience and expertise as one of the leading specialists in the field and this makes the book extremely readable. The material is chosen in such a way that it illuminates the key role of convexity in many areas of mathematics. Finally, it shows that convexity is a fundamental concept with a long history, a strong impact on other mathematical disciplines and an interior beauty, whilst still offering nontrivial open problems.