Exponential concentration in terms of Gromov-Ledoux's expansion coefficients on a metric measure space and its upper diameter bound satisfying volume doubling
type: Item
instance of: scholarly article
title: Exponential concentration in terms of Gromov-Ledoux's expansion coefficients on a metric measure space and its upper diameter bound satisfying volume doubling
zbMATH Open document ID: 1504.53058
author: Ushio Tanaka
published in: Osaka Journal of Mathematics
publication date: 2022-10-10T00:00:00Z
full work available at URL: 4846ojm.full
review text: The ``concentration of measure'' phenomenon on a metric measure space (equipped with a Borel probability measure) has many applications (see [\textit{M. Gromov} et al., 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. Boston, MA: Birkhäuser (1999; Zbl 0953.53002); \textit{M. Ledoux}, The concentration of measure phenomenon. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0995.60002)] and the references therein). The author discusses the relation between the observable diameter and the concentration function and proves a relevant inequality for the concentration function in terms of Gromov's and Ledoux's expansion coefficients (Theorem 5.5). For a bounded and doubling metric measure space he obtains an upper bound of the diameter in terms of Ledoux's expansion coefficients and the doubling constant (Theorem 7.1). Applying Theorem 7.1, the author extends the Bonnet-Myers Theorem to a closed smooth Riemannian manifold with non-nengative Ricci curvature and provides an upper bound for the diameter of the manifold in terms of the dimension and the spectral gap.
reviewed by: Jialong Deng
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 7598519
zbMATH Keywords: diameter
zbMATH Keywords: doubling metric measure spaces
zbMATH Keywords: concentration functions
zbMATH Keywords: Gromov's and Ledoux's expansion coefficients
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7598519