On the optimal weak transfer plan

We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show a link between Kantorovich duality and the permutation polytope by characterizing the transfer plan in two different ways. We prove that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a characterization for convex modified log sobolev inequality and a class of weak transport-entropy inequalities.

This talk contains results of a join work with N. Gozlan, C. Roberto, P-M. Samson and P. Tetali, and a join work of M. Strzelecki.