Sala Seminari (Dip. Matematica).
I will first give a short survey to recall the importance of convexity in the classical Monge problem. I will then introduce the $W_\infty$ Wasserstein distance and discuss the lack of convexity of the underlying problem. The next step will be to present the dual problem introduced in 2016 by Barron, Bocea and Jensen. The maximum value of this dual problem coincides with the minimum value of the transport problem but it is not clear that non trivial maximizers exist. I will finally show that in dimension 1 non trivial maximizers for the dual problem exist and allow to deduce a supplementary minimality property for the minimizers of the primal problem. (Joint work with Jean Louet).