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Two dimensional optimal transportation problem for a distance cost with a convex constraint

Ping ChenFeida JiangXiaoping Yang — 2013

ESAIM: Control, Optimisation and Calculus of Variations

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...

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