Convexity: An Analytic ViewpointConvexity 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
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 |
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Common terms and phrases
affine analytic Banach space bounded chapter characteristic function Choquet order compact convex subset concave functions cone continuous functions convergence theorem convex function convex set convex subset Corollary define Definition Let dense dfi(x discussed dual topology eigenvalues equimeasurable Example extreme points F is convex finite finite-dimensional given HLP theorem holds implies induction integral interval Krein-Milman theorem Legendre transforms Lemma Let f linear functional locally convex space Loewner Loewner's theorem log concave Math matrix matrix monotone maximal measure measure space monotone convergence theorem Moreover n x n neighborhood nonatomic nonempty nonnegative norm obeys open convex Orlicz spaces Pick positive definite Proof Let proof of Theorem Proposition prove Remark result Riesz Schur convex self-adjoint sequence shows simplex subspace suppose symmetric tangent theory topological vector space unique weak topology Young function zero