Asymptotic Theory Of Finite Dimensional Normed Spaces
Vol. 1200 of the LNM series deals with the geometrical structure of finite dimensional normed spaces. One of the main topics is the estimation of the dimensions of euclidean and lsubn p spaces which nicely embed into diverse finite-dimensional normed spaces. An essential method here is the concentration of measure phenomenon which is closely related to large deviation inequalities in Probability on the one hand, and to isoperimetric inequalities in Geometry on the other.\ The book contains...
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Vol. 1200 of the LNM series deals with the geometrical structure of finite dimensional normed spaces. One of the main topics is the estimation of the dimensions of euclidean and lsubn p spaces which nicely embed into diverse finite-dimensional normed spaces. An essential method here is the concentration of measure phenomenon which is closely related to large deviation inequalities in Probability on the one hand, and to isoperimetric inequalities in Geometry on the other. The book contains also an appendix, written by M. Gromov, which is an introduction to isoperimetric inequalities on riemannian manifolds. Only basic knowledge of Functional Analysis and Probability is expected of the reader. The book can be used (and was used by the authors) as a text for a first or second graduate course. The methods used here have been useful also in areas other than Functional Analysis (notably, Combinatorics).
Part I: The concentration of measure phenomenon in the theory of normed spaces. Preliminaries; The isoperimetric inequality on Ssub(n-1) and some consequences; Finite dimensional normed spaces; Almost euclidean subspaces of a normed space; Almost euclidian subspaces of l_psubn spaces of general n-dimensional normed spaces and of quotient of n-dimensional spaces; Levy families; Martingales; Embedding l_psubm into l_1subn; Type and cotype of normed spaces, and some simple relations with geometrical properties; Additional applications of Levy families in the theory of finite dimensional normed spaces. Part II: Type and cotype of normed spaces. Ramsey's theorem with some applications to normed spaces; Krivine's theorem; The Maurey-Pisier theorem; The Rademacher projection; Projections on random euclidean subspaces of finite dimensional normed spaces. Appendices:
