Duke Mathematical Journal

Existence of optimal transport maps for crystalline norms

L. Ambrosio, B. Kirchheim, and A. Pratelli

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Abstract

We show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in ℝn, assuming that the initial distribution of mass is absolutely continuous with respect to $\mathcal{L}$n. The proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the Euclidean distance as cost function. Moreover, improving a construction by Larman, we show the existence of a Nikodym set in ℝ3 having full measure in the unit cube, intersecting each element of a family of pairwise disjoint open lines only in one point. This example can be used to show that the regularity of the decomposition in transport rays plays an essential role in Sudakov-type arguments for proving the existence of optimal transport maps.

Article information

Source
Duke Math. J., Volume 125, Number 2 (2004), 207-241.

Dates
First available in Project Euclid: 27 October 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1098892268

Digital Object Identifier
doi:10.1215/S0012-7094-04-12521-7

Mathematical Reviews number (MathSciNet)
MR2096672

Zentralblatt MATH identifier
1076.49022

Subjects
Primary: 49J45: Methods involving semicontinuity and convergence; relaxation
Secondary: 28A50: Integration and disintegration of measures

Citation

Ambrosio, L.; Kirchheim, B.; Pratelli, A. Existence of optimal transport maps for crystalline norms. Duke Math. J. 125 (2004), no. 2, 207--241. doi:10.1215/S0012-7094-04-12521-7. https://projecteuclid.org/euclid.dmj/1098892268


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