Convexity: An Analytic Viewpoint

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Cambridge University Press, May 19, 2011 - Mathematics
Convexity is important in theoretical aspects of mathematics and also for economists and 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. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein–Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.
 

Contents

Convex functions and sets
1
Orlicz spaces
33
Gauges and locally convex spaces
51
Separation theorems
66
dual topologies bipolar sets and Legendre transforms
70
Monotone and convex matrix functions
87
a first proof
114
Extreme points and the KreinMilman theorem
120
Complex interpolation
185
The BrunnMinkowski inequalities and log concave functions
194
Rearrangement inequalities I BrascampLiebLuttinger inequalities
208
Rearrangement inequalities II Majorization
231
The relative entropy
278
Notes
287
References
321
Author index
339

The Strong KreinMilman theorem
136
existence
163
uniqueness
171

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About the author (2011)

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

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