This is a nice little book, providing a new look at the old subject of convexity and treating it from different points of view. As the authors suggest, parts of it can be used for a course on convexity for first year graduate students. The book consists of four chapters, the last of which is devoted to Choquet theory. Appendices present a necessary background on the separation of convex sets in LC spaces, the theory of semi-algebraic sets, applications to elliptic boundary value problems and the Horn conjecture concerning Weyl’s problem on spectra of the sum of Hermitean matrices. While elementary parts of the book can be presented even in a basic course of analysis, the book goes rather deeply into the subject. A unified approach based on an application of means of different types is combined with many classical and recent results to show to the reader convexity as one of central notions of mathematics. The book well documents sources of ideas and the origins of results and will be of interest even for specialists in the field; it may also be used as a reference book on the subject.