Exponential concentration in terms of Gromov-Ledoux's expansion coefficients on a metric measure space and its upper diameter bound satisfying volume doubling
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.
Gromov's and Ledoux's expansion coefficients
scientific article; zbMATH DE number 7598519
scientific article