Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces
Ambrosio, L., & Rajala, T. (2014). Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces. Annali di Matematica Pura ed Applicata, 193(1), 71-87. https://doi.org/10.1007/s10231-012-0266-x
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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2012. This is a final draft version of an article whose final and definitive form has been published by Springer. Published in this repository with the kind permission of the publisher.
We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier’s theorem of existence of optimal maps.
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