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Convexity
An Analytic Viewpoint

£72.00

Part of Cambridge Tracts in Mathematics

  • Author: Barry Simon, California Institute of Technology
  • Date Published: May 2011
  • availability: Available
  • format: Hardback
  • isbn: 9781107007314

£ 72.00
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  • 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.

    • The first monograph to comprehensively cover inequalities related to convexity
    • Covers a large number of topics unavailable in the monograph literature
    • Includes a detailed discussion of Choquet theory, essential for all analysts
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    Product details

    • Date Published: May 2011
    • format: Hardback
    • isbn: 9781107007314
    • length: 356 pages
    • dimensions: 231 x 157 x 25 mm
    • weight: 0.64kg
    • contains: 8 b/w illus.
    • availability: Available
  • Table of Contents

    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.

  • Author

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

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