Like all the books in the series Cambridge Tracts in Mathematics, this book “is devoted to a thorough, yet reasonably concise” treatment of a topic in mathematics, in this case the topic being Convexity. As usual, the Tract “takes up a single thread in a wide subject, and follows its ramifications, thus throwing light on its various aspects.” It is interesting to note that the series has another Tract on the subject: Convexity, by H. G. Eggleston, published in 1958. Although the first results on convexity were published by J. W. Gibbs in 1873, there have been many contributions since 1958, even very recent ones, all included in the impressive list (with almost 400 titles!) of References in the wonderful book by Barry Simon.
The book has sixteen chapters, beginning with the elementary definitions and results on convexity, and treating convex matrix functions, Loewner’s theorem, Krein-Milman theorem, Choquet theory, complex interpolation, Brunn-Minkowski inequalities, rearrangement inequalities, and a few other topics. The seventeenth chapter, Notes, discusses the history of convexity and comments on some of the topics discussed earlier in the book, proving to be very interesting in its own right. The book provides a very extensive treatment of all aspects of Convexity and its connections to other subjects, but (just when one thinks that these have been exhausted) the Notes end with a list of related issues that are not discussed previously.
The book is very well written and readable, has complete proofs and very good examples, which compensate the absence of exercises. Due to the connections emphasized throughout, mathematicians working in many areas can find useful results included in the text. Students can use (parts of) the book as well: motivated high school students and undergraduates can find some (elementary) results; graduate students can learn new things and enhance their research. Parts of the book could be used in teaching courses (not necessarily on convexity, but also in the many topics connected to it).
In summary, this is a wonderful book that should be part of the library of any mathematician and student of mathematics.
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.
1. Convex functions and sets
2. Orlicz spaces
3. Gauges and locally convex spaces
4. Separation theorems
5. Duality: dual topologies, bipolar sets, and Legendre transforms
6. Monotone and convex matrix functions
7. Loewner's theorem: a first proof
8. Extreme points and the Krein–Milman theorem
9. The strong Krein–Milman theorem
10. Choquet theory: existence
11. Choquet theory: uniqueness
12. Complex interpolation
13. The Brunn–Minkowski inequalities and log concave functions
14. Rearrangement inequalities: a) Brascamp–Lieb–Luttinger inequalities
15. Rearrangement inequalities: b) Majorization
16. The relative entropy