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**Asymptotic theory of finite dimensional normed spaces. With an appendix by M. Gromov: Isoperimetric inequalities in Riemannian manifolds.**
*(English)*
Zbl 0606.46013

Lecture Notes in Mathematics, 1200. Berlin etc.: Springer-Verlag. VIII, 156 p. DM 26.50 (1986).

This book deals with the geometrical structure of finite dimensional normed spaces as the dimension grows to infinity. This is a part of what came to be known as the local theory of Banach spaces. The purpose of this book is to introduce the reader to some of the results, problems and mainly methods developed in the local theory.

The contents of the book. Introduction. Part I: The concentration of measure phenomenon in the theory of normed spaces (1. Preliminaries; 2. The isoperimetric inequality on \(S^{n-1}\) and some consequences; 3. Finite dimensional normed spaces, preliminaries; 4. Almost Euclidean subspaces of a normed space; 5. Almost Euclidean subspaces of \(\ell^ n_ p\) spaces of general n-dimensional normed spaces, and of quotient of n-dimensional spaces; 6. Levy-families; 7. Martingales; 8. Embedding \(\ell^ m_ p\) into \(\ell^ n_ 1\); 9. Type and cotype of normed spaces, and some simple relations with geometrical properties; 10 Additional applications of Levy families in the theory of finite dimensional normed spaces).

Part II: Type and cotype of normed spaces (11. Ramsey’s theorem with some applications to normed spaces; 12. Krivine’s theorem; 13. The Maurey- Pisier theorem; 14. The Rademacher projection; 15. Projections on random Euclidean subspaces of finite dimensional normed spaces). Appendices (1. Isoperimetric inequalities in Riemannian manifolds, by M. Gromov; 2. Gaussian and Rademacher averages; 3. Kahane’s inequality; 4. Proof of the Beurling-Kato theorem 1.4; 5. The concentration of measure phenomenon for Gaussian variables).

The contents of the book. Introduction. Part I: The concentration of measure phenomenon in the theory of normed spaces (1. Preliminaries; 2. The isoperimetric inequality on \(S^{n-1}\) and some consequences; 3. Finite dimensional normed spaces, preliminaries; 4. Almost Euclidean subspaces of a normed space; 5. Almost Euclidean subspaces of \(\ell^ n_ p\) spaces of general n-dimensional normed spaces, and of quotient of n-dimensional spaces; 6. Levy-families; 7. Martingales; 8. Embedding \(\ell^ m_ p\) into \(\ell^ n_ 1\); 9. Type and cotype of normed spaces, and some simple relations with geometrical properties; 10 Additional applications of Levy families in the theory of finite dimensional normed spaces).

Part II: Type and cotype of normed spaces (11. Ramsey’s theorem with some applications to normed spaces; 12. Krivine’s theorem; 13. The Maurey- Pisier theorem; 14. The Rademacher projection; 15. Projections on random Euclidean subspaces of finite dimensional normed spaces). Appendices (1. Isoperimetric inequalities in Riemannian manifolds, by M. Gromov; 2. Gaussian and Rademacher averages; 3. Kahane’s inequality; 4. Proof of the Beurling-Kato theorem 1.4; 5. The concentration of measure phenomenon for Gaussian variables).

Reviewer: M.I.Kadets

### MSC:

46B25 | Classical Banach spaces in the general theory |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46B20 | Geometry and structure of normed linear spaces |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |