Two dimensional optimal transportation problem for a distance cost with a convex constraint
We first prove existence and uniqueness of optimal transportation maps for the Monge’s problem associated to a cost function with a strictly convex constraint in the Euclidean plane ℝ. The cost function coincides with the Euclidean distance if the displacement − belongs to a given strictly convex set, and it is infinite otherwise. Secondly, we give a sufficient condition for existence and uniqueness of optimal transportation maps for the original Monge’s problem in ℝ. Finally, we get existence...