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Klappentext 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. Zusammenfassung This book gives a self-contained presentation of some recent results! relating the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. This employs classical ideas from the theory of convex sets! probability theory! approximation theory and the local theory of Banach spaces. Inhaltsverzeichnis Introduction; 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; Bibliography; Index.
