Various properties of isoperimetric, functional, Transport-Entropy and
concentration inequalities are studied on a Riemannian manifold equipped with a
measure, whose generalized Ricci curvature is 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
$L^\infty$ bound on the ratio between their densities, Wasserstein distances,
and Kullback - Leibler divergence. In particular, an extension of the
Holley--Stroock perturbation lemma for the log-Sobolev inequality is obtained.
Second, the equivalence of Transport-Entropy inequalities with different cost
functions is verified, by obtaining a reverse Jensen type inequality. In view
of a recent result of Gozlan, this is used to obtain tensorization properties
of concentration inequalities with respect to various product-metrics, and the
tensorization result for isoperimetric inequalities of
Barthe--Cattiaux--Roberto is easily recovered. Some further applications are
also described. The main tool used is a previous precise result on the
equivalence between concentration and isoperimetric inequalities in the
described setting.