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December 2010 A strategy for non-strictly convex transport costs and the example of $║x-y║^p$ in $R^2$
Guillaume Carlier, Luigi De Pascale, Filippo Santambrogio
Commun. Math. Sci. 8(4): 931-941 (December 2010).

Abstract

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 existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form $h║x-y║$, with h strictly convex increasing and $║.║$ an arbitrary norm in $R^2$.

Citation

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Guillaume Carlier. Luigi De Pascale. Filippo Santambrogio. "A strategy for non-strictly convex transport costs and the example of $║x-y║^p$ in $R^2$." Commun. Math. Sci. 8 (4) 931 - 941, December 2010.

Information

Published: December 2010
First available in Project Euclid: 2 November 2010

zbMATH: 1226.49039
MathSciNet: MR2744914

Subjects:
Primary: 49J45 , 49K30 , 49Q20

Keywords: existence of optimal maps , general norms , Monge-Kantorovich problem , Optimal transport

Rights: Copyright © 2010 International Press of Boston

Vol.8 • No. 4 • December 2010
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