Two dimensional optimal transportation problem for a distance cost with a convex constraint
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1064-1075.

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 ℝ2. The cost function coincides with the Euclidean distance if the displacement y - x 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 ℝ2. Finally, we get existence of optimal transportation maps for a cost function with a convex constraint, i.e. y - x belongs to a given convex set with at most countable flat parts.

DOI : 10.1051/cocv/2013045
Classification : 49Q20, 49J45
Mots-clés : optimal transportation map, convex constraint, Monge transportation problem
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     title = {Two dimensional optimal transportation problem for a distance cost with a convex constraint},
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Chen, Ping; Jiang, Feida; Yang, Xiaoping. Two dimensional optimal transportation problem for a distance cost with a convex constraint. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1064-1075. doi : 10.1051/cocv/2013045. http://www.numdam.org/articles/10.1051/cocv/2013045/

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