Exponential Concentration in Terms of Gromov-Ledoux's Expansion Coefficients on a Metric Measure Space and Its Upper Diameter Bound Satisfying Volume Doubling

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.