Convexity is an ancient idea going back to Archimedes. Used sporadically in mathematical literature over the centuries, today it is a flourishing area of research. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.

This text, popular in its first edition, introduces analytic tools for studying convexity and provides analytical applications of the concept. It includes a general background on classical geometric theory, revealing a glimpse of how modern mathematics is developed and how geometric ideas may be studied analytically.

The book contains copious examples, many applications, and plenty of figures. It also includes an appendix which offers the technical tools needed to understand certain arguments in the book, a table of notation, and a thorough glossary to help readers with unfamiliar terms.

The book presents an analytic way to think about convexity theory. Although this means of doing things is well known to the experts, it is not well documented in the literature. The reader with only a basic background in real analysis (and perhaps a little linear algebra) can get a lot out of this book. This book is a definitive introductory text to the concept of convexity in the context of mathematical analysis and a suitable resource for students and faculty alike.