Abstract

In 1917, Paul Levy proved his classical isoperimetric inequality on the N-dimensional sphere. In the 1970's, Mikhail Gromov extended this inequality to all Riemannian manifolds with Ricci curvature bounded below by that of The N-sphere. Around the same time, the Concentration of Measure phenomenon was being put forth and studied by Vitali Milman. The relation between Concentration of Measure and Ricci curvature was realized shortly thereafter.
Elaborating on several articles, we begin by explicitly presenting a proof of the Concentration of Measure Inequality for the N-sphere as the archetypical space of positive curvature, followed by a complete proof extending this result to all Riemannian manifolds with Ricci curvature bounded below by that of the N-sphere in the process, we present a detailed technical proof of the Gromov-Levy isoperimetric inequality.
Following Yann Ollivier, we note and prove a Concentration of Measure inequality on the discrete Hamming cube, and discuss his extension of Ricci curvature to general metric spaces, particularly discrete metric measure spaces. We show that this “coarse” Ricci curvature on the Hamming cube is positive and present Ollivier's Concentration of Measure inequality for all spaces admitting positive coarse Ricci curvature. In addition, we calculate the coarse Ricci curvature for several discrete metric spaces.