Geodesic
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

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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
Keywords: 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},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1064--1075},
     publisher = {EDP-Sciences},
     volume = {19},
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     year = {2013},
     doi = {10.1051/cocv/2013045},
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     zbl = {1282.49036},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013045/}
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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

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