Concentration, results and applications.

*(English)*Zbl 1057.46011
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1603-1634 (2003).

This survey is devoted to concentration phenomena, approximate isoperimetric inequalities, and their applications in the asymptotic theory of finite-dimensional spaces.

The introduction contains definitions and motivations. In Section 2, the author presents many inequalities for various probability spaces and some geometric inequalities, and discusses the methods of their proofs. In particular, the following inequalities are presented: Brunn-Minkowski inequality, Prékopa-Leindler inequality, Lévy’s isoperimetric inequality on the Euclidean sphere, Talagrand’s concentration inequalities on product spaces, logarithmic Sobolev inequality. Special attention is paid to martingales and martingale technique.

Section 3 deals with applications. As examples of use of the concentration of Lebesgue measure on the Euclidean sphere, the author presents Milman’s proof of Dvoretzky’s theorem and a proof of the Figiel-Lindenstrauss lemma. He then turns to fine embeddings of subspaces of \(L_p\) in \(\ell _p^n\) and discusses generalizations of the result of Johnson and Schechtman that \(\ell _p^{cn}\) nicely embeds in \(\ell _1^n\). Finally, a construction of symmetric block bases in finite-dimensional normed spaces is given.

For the entire collection see [Zbl 1013.46001].

The introduction contains definitions and motivations. In Section 2, the author presents many inequalities for various probability spaces and some geometric inequalities, and discusses the methods of their proofs. In particular, the following inequalities are presented: Brunn-Minkowski inequality, Prékopa-Leindler inequality, Lévy’s isoperimetric inequality on the Euclidean sphere, Talagrand’s concentration inequalities on product spaces, logarithmic Sobolev inequality. Special attention is paid to martingales and martingale technique.

Section 3 deals with applications. As examples of use of the concentration of Lebesgue measure on the Euclidean sphere, the author presents Milman’s proof of Dvoretzky’s theorem and a proof of the Figiel-Lindenstrauss lemma. He then turns to fine embeddings of subspaces of \(L_p\) in \(\ell _p^n\) and discusses generalizations of the result of Johnson and Schechtman that \(\ell _p^{cn}\) nicely embeds in \(\ell _1^n\). Finally, a construction of symmetric block bases in finite-dimensional normed spaces is given.

For the entire collection see [Zbl 1013.46001].

Reviewer: A. E. Litvak (Edmonton)

##### MSC:

46B07 | Local theory of Banach spaces |

46-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis |

60-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to probability theory |

28A35 | Measures and integrals in product spaces |

46B09 | Probabilistic methods in Banach space theory |

46B20 | Geometry and structure of normed linear spaces |

52A21 | Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) |

60B11 | Probability theory on linear topological spaces |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |