Various properties of isoperimetric, Sobolev-type and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is (possibly negatively) bounded from below.

First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided L8 bound on the ratio between their densities, Total-Variation, Wasserstein distances, and relative entropy (Kullback-Leibler divergence). In particular, an extension of the Holley-Stroock perturbation lemma for the log-Sobolev inequality is obtained, and the dependence on the perturbation parameter is improved from linear to logarithmic.

Next, in the compact setting, an optimal (up to numeric constants) isoperimetric inequality is obtained as a function of the curvature lower bound and diameter upper bound. In particular, the best known log-Sobolev inequality is obtained in this setting.

Time permitting, we will also mention the equivalence of Transport-Entropy inequalities with different cost functions and some of their applications.

The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting.