## Asymptotic Theory of Finite Dimensional Normed Spaces: Isoperimetric Inequalities in Riemannian ManifoldsThis 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 (this name was derived from the fact that in its first stages, this theory dealt mainly with relating the structure of infinite dimensional Banach spaces to the structure of their lattice of finite dimensional subspaces). Our purpose in this book is to introduce the reader to some of the results, problems, and mainly methods developed in the Local Theory, in the last few years. This by no means is a complete survey of this wide area. Some of the main topics we do not discuss here are mentioned in the Notes and Remarks section. Several books appeared recently or are going to appear shortly, which cover much of the material not covered in this book. Among these are Pisier's [Pis6] where factorization theorems related to Grothendieck's theorem are extensively discussed, and Tomczak-Jaegermann's [T-Jl] where operator ideals and distances between finite dimensional normed spaces are studied in detail. Another related book is Pietch's [Pie]. |

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### Contents

1 | |

The isoperimetric inequality on S and some consequences | 9 |

Almost euclidean subspaces of t Spaces | 19 |

Levy families | 27 |

Martingales | 42 |

Type and cotype of normed spaces and some | 51 |

Additional applications of Levy families | 60 |

Type and cotype of normed spaces | 69 |

The Rademacher projection | 98 |

Projections on random euclidean subspaces | 106 |

Isoperimetric inequalities in Riemannian Manifolds | 114 |

Gaussian and Rademacher averages | 130 |

Proof of the BeurlingKato Theorem 14 4 | 137 |

Notes and Remarks | 144 |

151 | |

153 | |

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1-unconditional absolute constant assume Banach space block finitely representable Borel boundary C°-smooth Ca(X Chapter compact constants c1 convex corollary define deformation denote diffeomorphically dimensional normed spaces dimensional subspace Dvoretzky's Theorem eigenvalue equals estimate Euclidean structure example finite dimensional normed finite sequences follows function f gaussian Haar measure hyperbolic plane hypersurface implies integer interior normal isometric isoperimetric inequality J(vo Jacobian Kahane's inequality Lemma Let F Levy’s linear Lipschitz map F mean curvature median metric space n-dimensional normal geodesic map normal Levy family normed space open subset operator principal curvatures Proposition prove Rademacher functions radius random variables resp Ricci curvature Riemannian manifolds Riemannian metric Riemannian structure satisfies scalars smooth submanifold subspace tangent Theorem 4.2 To(V triangle inequality type and cotype unit ball unit vector basis volume