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

Ping Chen; Feida Jiang; Xiaoping Yang

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 4, page 1064-1075
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Chen, Ping, Jiang, Feida, and Yang, Xiaoping. "Two dimensional optimal transportation problem for a distance cost with a convex constraint." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 1064-1075. <http://eudml.org/doc/272802>.

@article{Chen2013,
abstract = {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.},
author = {Chen, Ping, Jiang, Feida, Yang, Xiaoping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal transportation map; convex constraint; Monge transportation problem},
language = {eng},
number = {4},
pages = {1064-1075},
publisher = {EDP-Sciences},
title = {Two dimensional optimal transportation problem for a distance cost with a convex constraint},
url = {http://eudml.org/doc/272802},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Chen, Ping
AU - Jiang, Feida
AU - Yang, Xiaoping
TI - Two dimensional optimal transportation problem for a distance cost with a convex constraint
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 4
SP - 1064
EP - 1075
AB - 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.
LA - eng
KW - optimal transportation map; convex constraint; Monge transportation problem
UR - http://eudml.org/doc/272802
ER -

References

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