Metric measure geometry. Gromov's theory of convergence and concentration of metrics and measures
type: Item
instance of: scholarly article
title: Metric measure geometry. Gromov's theory of convergence and concentration of metrics and measures
zbMATH Open document ID: 1335.53003
DOI: 10.4171/158
published in: IRMA Lectures in Mathematics and Theoretical Physics
publication date: 2016-01-07T00:00:00Z
review text: This book is dedicated to the study, completion and generalization of the theory of metric geometry on the space of metric measure spaces, originally developed by \textit{M. Gromov} in his book [Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. 3rd printing. Modern Birkhäuser Classics. Basel: Birkhäuser (2007; Zbl 1113.53001)] which in turn is based on the idea of the concentration of measure phenomena due to \textit{P. Lévy} [Problèmes concrets d'analyse fonctionnelle. - Seconde éd. des Leçons d'analyse fonctionnelle. - Avec un complément par F. Pellegrino. Paris: Gauthier-Villars (1951; Zbl 0043.32302)] and \textit{V. D. Mil'man} [Funct. Anal. Appl. 5, 288--295 (1972); translation from Funkts. Anal. Prilozh. 5, No. 4, 28--37 (1971; Zbl 0239.46018)] and [in: Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157--158, 273--301 (1988; Zbl 0681.46021)]. {Contents.} Introduction, 1. Prelimimaries from measure theory, 2. The Lévy-Milman concentration phenomenon, 3. Gromov-Hausdorff distance and distance matrix, 4. Box distance, 5. Observable distance and measurement, 6. The space of pyramids, 7. Asymptotic concentration, 8. Dissipation, 9. Curvature and concentration, Bibliography, Index. The author's new theory is based of the notion of observable distance. A \textit{metric measure space}, short \textit{mm-space}, is a triple \((X,d_X,\mu_X)\), where \((X,d_X)\) is a complete separable metric space and \(\mu_X\) is a Borel probability measure of \(X\). A sequence of mm-spaces \(X_n\), \(n=1,2,\dots\), is called a \textit{Lévy family} if \(\underset{n\rightarrow \infty }{\lim }\underset{c\in \mathbb R}{\inf}\mu_{X_n}(| f_n-c|>\varepsilon)=0\), for any sequence of 1-Lipschitz continuous functions \(f_n:X_n\rightarrow \mathbb R\), \(n=1,2,\dots\), and for any \(\varepsilon >0\). Example: the sequence…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 60B10
zbMATH DE Number: 6527202
zbMATH Keywords: Alexandrov space (curvature)
zbMATH Keywords: asymptotic concentration
zbMATH Keywords: Borel measure
zbMATH Keywords: box distance
zbMATH Keywords: concentration pyramid
zbMATH Keywords: function
zbMATH Keywords: topology
zbMATH Keywords: dissipation
zbMATH Keywords: distance matrix
zbMATH Keywords: Gaussian measure space
zbMATH Keywords: geodesic space
zbMATH Keywords: Gromov-Hausdorff distance
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: Ky Fan metric
zbMATH Keywords: Lévy family
zbMATH Keywords: Lipschitz order
zbMATH Keywords: Maxwell-Boltzmann distribution law
zbMATH Keywords: measured-Gromov-Hausdorff convergence
zbMATH Keywords: metric measure space
zbMATH Keywords: mm-isomorphism
zbMATH Keywords: observable distance
zbMATH Keywords: parameter
zbMATH Keywords: pseudo-metric
zbMATH Keywords: pyramid
zbMATH Keywords: spectral compactness
zbMATH Keywords: spectral concentration
zbMATH Keywords: standard Gaussian measure
zbMATH Keywords: weak Hausdorff convergence
author: Takashi Shioya
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/158
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description: scientific article; zbMATH DE number 6527202