Convex and discrete geometry / Peter M. Gruber

This excellent book presents a complete and coherent overview of principal ideas, results and methods of convex and discrete geometry. The text consists of four chapters: 1. Convex functions; 2. Convex bodies; 3. Convex polytopes; 4. Geometry of numbers and aspects of discrete geometry. The topics covered include theory of convex functions of one and several variables, mixed volumes, Brunn-Minkowski inequality, isoperimetric inequalities, symmetrization and approximation of convex bodies, combinatorial and metric properties of convex polytopes, lattice polytopes and linear optimization, packing of convex bodies, covering with convex bodies, tiling with convex polytopes.

The writing is really fascinating, the author provides a clear and accurate exposition of fundamental results on convexity. The style of presentation is chosen in such a way that the reader may really enjoy a panoramic view of the convex geometry. Some statements and proofs are explained quite rigorously, whereas another ones are rather concise and present central ideas without technical details. Discussions are often followed by appropriate references for further readings. To make the book more lively, the author includes amusing historical remarks together with numerous citations of famous mathematicians. The list of carefully selected references includes more than one thousand items.

The book is highly recommended to everyone who is interested in convex geometry: it may be viewed as a graduate-level introduction as well as a source of both classical and actual results in the subject. (Zentralblatt)