Volume 19, Number 4, October-December 2013
|Page(s)||1064 - 1075|
|Published online||04 July 2013|
Two dimensional optimal transportation problem for a distance cost with a convex constraint∗
School of Science, Nanjing University of Science and
firstname.lastname@example.org; email@example.com; firstname.lastname@example.org
2 School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, P.R. China
Received: 11 May 2012
Revised: 30 December 2012
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.
Mathematics Subject Classification: 49Q20 / 49J45
Key words: Optimal transportation map / convex constraint / Monge transportation problem
© EDP Sciences, SMAI, 2013
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