This thesis is devoted to the study of some applications of the so-called localization technique in the setting of compact Riemannian manifolds. The first appearance of this technique, essentially based on the reduction of an n-dimensional problem to a family of one dimensional problems, dates back to a work of Payne and Weinberger, where the authors proved a Poincaré-type inequality on convex domains of the Euclidean space by means of an iterative dimension reduction procedure.

We approach two different problems via localization techniques. The first one is the Monge transport problem for distance cost on Riemannian manifolds. The second one is the Lévy-Gromov isoperimetric inequality (and its infinite dimensional counterpart, namely the Bakry-Ledoux inequality). The bridge between these two results is given by the study of the localization properties of the so-called curvature-dimension condition.