Convexity

Convexity provides a wide-ranging introduction for final year undergraduates and graduate students. Convex sets and functions are studied in the Euclidean space IRn, thus allowing an exposition demanding only an elementary knowledge of analysis and linear algebra, and enabling concepts to be motivated through simple geometric examples.

The fundemental ideas of convexity are natural and appealing, and does not have to travel far along its path, before meeting significant, aesthetically pleasing results. It develops geometric intuition, and is a showcase for displaying interconnections amongst different parts of mathematics, in addition to have ties with economics, science and engineering. Despite being an active research field, it abounds in unsolved problems having an instant intuitive appeal.

One distinctive feature of the book is the diverse applications that it highlights: number theory, geometric extremum problems, combinatorial geometry, linear programming, game theory, polytopes, bodies of constant width, the gamma function, minimax approximation, and linear, classical and matrix inequalities. Several topics make their first appearance in a general introduction to convexity, while a few have not appeared outside research journals. The account has a self-contained treatment of volume, thus permitting a rigorous discussion of mixed volumes, is operimetry and Brunn-Minkowski theory. Full solutions to most of the 241 exercises are provided and detailed suggestions for further reading are given.