Gromov's Problem: Bound the Expansion Coefficient from below in terms of the Observable Diameter of a Metric Measure Space, and its Diameter Bounds

Abstract

In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from below in terms of the observable diameter. The overall aim of the current study is to demonstrate the answer to this problem. To begin solving this problem, the concentration of measure phenomenon on the metric measure space must be considered. The concentration function to evaluate the measure phenomenon is connected by the observable diameter and the expansion coefficient. Furthermore, the procedure for our answer gives us the upper bound for the expansion coefficient in terms of the observable diameter. Combining the desired lower bound for the expansion coefficient with its upper bound, we eventually obtain the upper bound for the observable diameter. Simultaneously, this reasoning has enabled us to obtain the upper bound for the diameter of a bounded metric measure space in terms of the expansion coefficient. We will apply the above-mentioned results to a compact connected Riemannian manifold with non-negative Ricci curvature, which makes the bounds more explicit. More precisely, they are in terms of the doubling constant of the Riemannian measure and the first non-trivial eigenvalue of the Laplacian on the Riemannian manifold.