Convexity
Subtitle: an analytic viewpoint
By: Barry Simon
Series: Cambridge tracts in mathematics -- 187; Cambridge tracts in mathematics -- 187.
Publisher: Cambridge University Press
Published: 2011
Place of publication: Camrbridge, UK; New York
Language: eng
Number of pages: 345
ISBN-13: 9781107007314
Subjects: Convex domains; Mathematical analysis
Description: "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"--
Notes: Includes bibliographical references (p. [321]-338) and index.
Table of contents: title: Convex functions and sets; title: Orlicz spaces; title: Gauges and locally convex spaces; title: Separation theorems; title: Duality : dual topologies, bipolar sets, and Legendre transforms; title: Monotone and convex matrix functions; title: Loewner's theorem : a first proof; title: Extreme points and the Krein-Milman theorem; title: The strong Krein-Milman theorem; title: Choquet theory : existence; title: Choquet theory : uniqueness; title: Complex interpolation; title: The Brunn-Minkowski inequalities and log concave functions; title: Rearrangement inequalities. Brascamp-Lieb-Luttinger inequalities ; Rearrangement inequalities; title: Majorization. The relative entropy.