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A strategy for non-strictly convex transport costs and the example of ║x−y║p in R2

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Date
2010
Link to item file
https://hal.archives-ouvertes.fr/hal-00417303/
Dewey
Analyse
Sujet
optimal transport; Monge-Kantorovich problem; existence of optimal maps; general norms
Journal issue
Communications in Mathematical Sciences
Volume
8
Number
4
Publication date
2010
Article pages
931-941
Publisher
International Press
DOI
http://dx.doi.org/10.4310/CMS.2010.v8.n4.a8
URI
https://basepub.dauphine.fr/handle/123456789/3518
Collections
  • CEREMADE : Publications
Metadata
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Author
Carlier, Guillaume
Santambrogio, Filippo
de Pascale, Luigi
Type
Article accepté pour publication ou publié
Abstract (EN)
This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As a consequence of our procedure, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. To illustrate possible results obtained through this general approach, we prove exisence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesque measure and the transportation cost is of the form h(\| x-y\|) with h strictly convex increasing and \|.\| an arbitrary norm in R^2.

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