Discrete Ricci curvature with applications

We define a notion of discrete Ricci curvature for a metric measure space by looking at whether “small balls are closer than their centers are”. In a Riemannian manifolds this gives back usual Ricci curvature up to scaling. This definition is very easy to apply in a series of examples such as graphs (eg the discrete cube has positive curvature). We are able to generalize several Riemannian theorems in positive curvature, such as concentration of measure and the log-Sobolev inequality. This definition also allows to prove new theorems both in the Riemannian and discrete.