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October 2022 Exponential Concentration in Terms of Gromov-Ledoux's Expansion Coefficients on a Metric Measure Space and Its Upper Diameter Bound Satisfying Volume Doubling
Ushio Tanaka
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Osaka J. Math. 59(4): 727-753 (October 2022).

Abstract

To investigate a concentration of measure phenomena on metric measure spaces in terms of Gromov-Ledoux's expansion coefficients on this space as well as Ledoux's per se, we studied a concentration function in concert with their expansion coefficients. Further investigation into an exponential concentration in terms of Ledoux's expansion coefficient on a bounded and volume doubling metric measure space enables us to derive an upper bound for its diameter in terms of both the Ledoux's expansion coefficient and doubling constant, provided that Ledoux's expansion coefficient $> 1$. In this study, we let Ledoux's expansion coefficient $> 1$ on a metric measure space, which is ensured by adopting Poincaré inequality. We demonstrated that on a metric measure space, Gromov-Ledoux's expansion coefficients with Ledoux's expansion coefficient $> 1$ give rise to an exponential concentration in terms of themselves. We further showed that on a bounded and volume doubling metric measure space, a Ledoux's expansion coefficient of order bounded from above in terms of both the doubling constant $> 1$ and its diameter is bounded from above in terms of the doubling constant per se. We applied this upper diameter bound to a closed smooth Riemannian manifold with non-negative Ricci curvature. This upper bound is described in terms of both the spectral gap and dimension.

Acknowledgments

The present author thanks to S. Ohta, a professor of Osaka University; T. Yokota, an associate professor of Tohoku University; Y. Kitabeppu, an associate professor of Kumamoto University; R. Ozawa, a lecturer of National Defense Academy for their fruitful comments on an earlier version of this article. The author enjoyed arguments over the present study with them through the seminar constantly convened at Kyoto University. Furthermore, he extends his appreciation to Prof. S. Ohta, who accepted his offer to criticise its revised version; in particular, an observation of Propositions \ref{prop:monotone properties of Gromov's expansion coefficient} and \ref{prop:monotone properties of Ledoux's expansion coefficient} is due to him. The present paper owes its existence to the seminar with them. The author is grateful to a referee for some useful comments.\par

This work was partially supported by Grant-in-Aid for Young Scientists (B), JSPS KAKENHI Grant Number JP25730022.

Citation

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Ushio Tanaka. "Exponential Concentration in Terms of Gromov-Ledoux's Expansion Coefficients on a Metric Measure Space and Its Upper Diameter Bound Satisfying Volume Doubling." Osaka J. Math. 59 (4) 727 - 753, October 2022.

Information

Received: 1 May 2018; Revised: 18 February 2021; Published: October 2022
First available in Project Euclid: 4 October 2022

Subjects:
Primary: 53C23
Secondary: 51F99

Rights: Copyright © 2022 Osaka University and Osaka Metropolitan University, Departments of Mathematics

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Vol.59 • No. 4 • October 2022
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