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.
A comprehensive look at convexity and its mathematical ramifications.
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
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