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In the celebrated book entitled Metric Structures for Riemannian andNon-Riemannian Spaces, so-called Green Book, Gromov presented a problemregarding a metric measure space. Gromov posed the question Bound the expansioncoefficient from below in terms of the observable diameter. The overall aim ofthe current study is to demonstrate the answer to this problem. To beginsolving this problem, the concentration of measure phenomenon on the metricmeasure space must be considered. The concentration function to evaluate themeasure phenomenon is connected by the observable diameter and the expansioncoefficient. Furthermore, the procedure for our answer gives us the upper boundfor the expansion coefficient in terms of the observable diameter. Combiningthe desired lower bound for the expansion coefficient with its upper bound, weeventually obtain the upper bound for the observable diameter. Simultaneously,this reasoning has enabled us to obtain the upper bound for the diameter of abounded metric measure space in terms of the expansion coefficient. We willapply the above-mentioned results to a compact connected Riemannian manifoldwith non-negative Ricci curvature, which makes the bounds more explicit. Moreprecisely, they are in terms of the doubling constant of the Riemannian measureand the first non-trivial eigenvalue of the Laplacian on the Riemannianmanifold.

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