Abstract
We study the nonlinear optimal transportation problem of minimizing the functional \(C(\lambda )={{\mathrm{\lambda -ess\,sup}}}c\) among transport plans with given marginals. We present some general results regarding the problem, particularly connecting “good” solutions to a suitable definition of cyclical monotonicity. We show that cyclically monotone transport plans are induced by transport maps in \(\mathbb {R}^n\) under relatively general assumptions on the first marginal and the cost function. With additional assumptions we are also able to prove results about continuity and uniqueness of these optimal maps.
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Acknowledgments
The author would like to thank Petri Juutinen for introducing the author to the \(L^\infty \) mass transport problem, Tapio Rajala for sharing his thoughts about the assumptions on the first marginal and how this relates to the existence of optimal transport maps, and finally Antti Käenmäki for his help when the author tried to understand Mr. Rajala’s ideas. The author has been supported by University of Jyväskylä and the Vilho, Yrjö and Kalle Väisälä foundation.
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Jylhä, H. The \(L^\infty \) optimal transport: infinite cyclical monotonicity and the existence of optimal transport maps. Calc. Var. 52, 303–326 (2015). https://doi.org/10.1007/s00526-014-0713-1
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DOI: https://doi.org/10.1007/s00526-014-0713-1