  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.
