Convex Functions and Their Applications

This second edition provides a thorough introduction to contemporary convex function theory with many new results. A large variety of subjects are covered, from the one real variable case to some of the most advanced topics. The new edition includes considerably more material emphasizing the rich applicability of convex analysis to concrete examples. Chapters 4, 5, and 6 are entirely new, covering important topics such as the Hardy-Littlewood-Pólya-Schur theory of majorization, matrix convexity, and the Legendre-Fenchel-Moreau duality theory. This book can serve as a reference and source of inspiration to researchers in several branches of mathematics and engineering, and it can also be used as a reference text for graduate courses on convex functions and applications.

Convex Functions on Intervals.- Convex Sets in Real Linear Spaces.- Convex Functions on a Normed Linear Space.- Convexity and Majorization.- Convexity in Spaces of Matrices.- Duality and Convex Optimization.- Special Topics in Majorization Theory.- A. Generalized Convexity on Intervals.- B. Background on Convex Sets.- C. Elementary Symmetric Functions.- D. Second Order Differentiability of Convex Functions.- E. The Variational Approach of PDE.