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The Volume of Convex Bodies and Banach Space Geometry

The Volume of Convex Bodies and Banach Space Geometry

Part of Cambridge Tracts in Mathematics

  • Author: Gilles Pisier, Université de Paris VI (Pierre et Marie Curie)
  • Date Published: May 1999
  • availability: Available
  • format: Paperback
  • isbn: 9780521666350


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About the Authors
  • This book aims to give a self-contained presentation of a number of results, which relate the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. The methods employ classical ideas from the theory of convex sets, probability theory, approximation theory and the local theory of Banach spaces. The book is in two parts. The first presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. The second part gives a detailed exposition of the recently introduced classes of Banach spaces of weak cotype 2 or weak type 2, and the intersection of the classes (weak Hilbert space). The book is based on courses given in Paris and in Texas.

    • Can be used for a graduate course
    • Has sold over 1100 copies in hardback
    • Based on courses taught by the author
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    Product details

    • Date Published: May 1999
    • format: Paperback
    • isbn: 9780521666350
    • length: 268 pages
    • dimensions: 228 x 153 x 16 mm
    • weight: 0.365kg
    • availability: Available
  • Table of Contents

    1. Notation and preliminary background
    2. Gaussian variables. K-convexity
    3. Ellipsoids
    4. Dvoretzky's theorem
    5. Entropy, approximation numbers, and Gaussian processes
    6. Volume ratio
    7. Milman's ellipsoids
    8. Another proof of the QS theorem
    9. Volume numbers
    10. Weak cotype 2
    11. Weak type 2
    12. Weak Hilbert spaces
    13. Some examples: the Tsirelson spaces
    14. Reflexivity of weak Hilbert spaces
    15. Fredholm determinants
    Final remarks

  • Author

    Gilles Pisier, Université de Paris VI (Pierre et Marie Curie)

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