Asymptotic Theory of Finite Dimensional Normed Spaces: Isoperimetric Inequalities in Riemannian Manifolds (Google eBook)
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 l^n 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).
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absolute constant assume Banach space block finitely representable Borel Chapter compact convex corollary cotype q constant define denote dimensional normed spaces dimensional subspace Dvoretzky's Theorem ellipsoid estimate euclidean norm Euclidean structure example exists a constant family with constants finite dimensional normed finite sequences follows gaussian given Haar measure hypersurface implies interior normal invariant isometric isoperimetric inequality Kahane's inequality Lemma linear Lipschitz map F martingale Maurey mean curvature median metric space n-dimensional normal Levy family normalized Haar measure Note operator orthogonal orthonormal Pisier probability space projection proof of Theorem Proposition prove Rademacher functions radius Ramsey's Theorem random variables resp Riemannian manifolds Riemannian metric satisfies scalars semigroup Sn_1 Sn+1 sphere subset subspace symmetric p-stable tangent Theorem 4.2 triangle inequality TV(V type and cotype unit ball unit vector basis volume zero