The new book Convex Functions by Jonathan M. Borwein and Jon D. Vanderwerff has been selected as one of the “Outstanding Academic Titles” for 2011 by Choice, the American Library Association’s library book review journal.
Here is an excerpt from a review written by John D. Cook and published by the Mathematical Association of America in their Mathematical Sciences Digital Library:
When mathematicians say a function is “nonlinear” they often mean that it is not necessarily linear. In this sense “nonlinear” is not an assumption but rather the absence of an assumption. To make progress in studying a nonlinear problem, we have to make some assumption about how a function departs from linearity. We have to replace an assumption of linearity with a weaker assumption that still retains enough structure to allow us to prove theorems. Often that weaker assumption is convexity. In large-scale optimization, for example, convexity is just the right assumption in order to retain many of the benefits of the linear theory while greatly increasing its scope of application. The study of convex functions has become more popular as nonlinear problems have become more popular and researchers realize they need to assume a particular kind of nonlinearity. …
Convex Functions tells a story from beginning to end. It starts with examples of convex functions in order to motivate the reader. It then progresses further and further into the theory, introducing special cases before proceeding to more general theory. The book closes with a retrospective, revisiting the differences between convex functions over finite and infinite dimensional spaces. The authors introduce a small amount of redundancy to make the book easier to read.