Convexity

Informationen zum Autor Barry Simon is IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. Klappentext A comprehensive look at convexity and its mathematical ramifications. Zusammenfassung Convexity is important for theoretical aspects of mathematics and also for economists and theoretical 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. Inhaltsverzeichnis Preface; 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.

Klappentext

A comprehensive look at convexity and its mathematical ramifications.

Zusammenfassung
Convexity is important for theoretical aspects of mathematics and also for economists and theoretical 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.

Inhalt

Preface; 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.