The Volume of Convex Bodies and Banach Space Geometry

by Gilles Pisier

• ISBN13: 9780521666350
• ISBN10: 052166635X
• Format: Paperback
• Copyright: 1999-06-13
• Publisher: Cambridge University Press

Summary

Now in paperback, this popular book gives a self-contained presentation of a number of recent 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 first part of the book presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. In the second part Pisier 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). This text will be a superb choice for courses in analysis and probability theory.

Table of Contents

Introduction

1. Notation and preliminary background

2. Gaussian variable, 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

Bibliography

Index.