Barry Simon is IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology.

"Simon's monograph is a valuable addition to the literature on convexity that will inspire many minds enchanted by the beauty and power of the cornerstone of functional analysis." S. Kutateladze, Mathematical Reviews

• 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.

Barry Simon is IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology.

"Simon's monograph is a valuable addition to the literature on convexity that will inspire many minds enchanted by the beauty and power of the cornerstone of functional analysis." S. Kutateladze, Mathematical Reviews

• 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.

Barry Simon is IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology.

"Simon's monograph is a valuable addition to the literature on convexity that will inspire many minds enchanted by the beauty and power of the cornerstone of functional analysis." S. Kutateladze, Mathematical Reviews

• 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.