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Transition to nestedness in multi- to one-dimensional optimal transport



We study a one-parameter class of examples of optimal transport problems between a two-dimensional source and a one-dimensional target. Our earlier work identified a nestedness condition on the surplus function and marginals, under which it is possible to solve the problem semi-explicitly. In the family of examples we consider, we classify the values of parameters which lead to nestedness. In those cases, we derive an almost explicit characterisation of the solution.



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[1]Chiappori, P.-A., McCann, R. J. & Pass, B. (submitted, 2016) Multidimensional matching. Preprint at arXiv:1604.05771.
[2]Chiappori, P.-A., McCann, R. J. & Pass, B. (2017) Multi- to one-dimensional optimal transport. Commun. Pure Appl. Math. 70(12), 24052444.
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[4]McCann, R. J. & Pass, B. (submitted, 2018) Optimal transportation between unequal dimensions. Preprint at
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[6]Santambrogio, F. (2015) Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling. Progress in Nonlinear Differential Equations and their Applications, Vol. 87, Birkhäuser/Springer, Cham.
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[8]Villani, C. (2009) Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer, New York.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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