Convexity an analytic viewpoint
Convex functions and sets -- Orlicz spaces -- Gauges and locally convex spaces -- Separation theorems -- Duality : dual topologies, bipolar sets, and Legendre transforms -- Monotone and convex matrix functions -- Loewner's theorem : a first proof -- Extreme points and the Krein-Milman theorem -...
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| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
Cambridge, New York, Melbourne
Cambridge University Press
2011
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| Schriftenreihe: | Cambridge tracts in mathematics
187 |
| Schlagworte: | |
| Online Zugang: | Inhaltsverzeichnis Inhaltstext |
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Tag hinzufügen | Zusammenfassung: | Convex functions and sets -- Orlicz spaces -- Gauges and locally convex spaces -- Separation theorems -- Duality : dual topologies, bipolar sets, and Legendre transforms -- Monotone and convex matrix functions -- Loewner's theorem : a first proof -- Extreme points and the Krein-Milman theorem -- The strong Krein-Milman theorem -- Choquet theory : existence -- Choquet theory : uniqueness -- Complex interpolation -- The Brunn-Minkowski inequalities and log concave functions -- Rearrangement inequalities. Brascamp-Lieb-Luttinger inequalities ; Rearrangement inequalities -- Majorization. The relative entropy. "Convexity of sets and functions are extremely simple notions to define, so it may be somewhat surprising the depth and breadth of ideas that these notions give rise to. It turns out that convexity is central to a vast number of applied areas, including Statistical Mechanics, Thermodynamics, Mathematical Economics, and Statistics,and that many inequalities, including Hl̲der's and Minkowski's inequalities, are related to convexity. An introductory chapter (1) includes a study of regularity properties of convex functions, some inequalities (Hl̲der, Minkowski, and Jensen), the Hahn-Banach theorem as a statement about extending tangents to convex functions, and the introduction of two constructions that will play major roles later in this book: the Minkowski gauge of a convex set and the Legendre transform of a function"-- |
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| Beschreibung: | Literaturverzeichnis Seite [321]-338 Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | ix, 345 Seiten Illustrationen |
| ISBN: | 1107007313 1-107-00731-3 9781107007314 978-1-107-00731-4 |