  We develop the optimal transportation approach to modified log-Sobolev
inequalities and to isoperimetric inequalities. Various sufficient conditions
for such inequalities are given. Some of them are new even in the classical
log-Sobolev case. The idea behind many of these conditions is that measures
with a non-convex potential may enjoy such functional inequalities provided
they have a strong integrability property that balances the lack of convexity.
In addition, several known criteria are recovered in a simple unified way by
transportation methods and generalized to the Riemannian setting.
