Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein-Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.
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Table of ContentsPreface; 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; 17. Notes; References; Author index; Subject index.