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Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis

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  • In this survey paper I describe the convoluted links between theregularity theory of optimal transport and the geometry of cut locus.
    Mathematics Subject Classification: 35J60, 53A05.

    Citation:

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  • [1]

    M. Castelpietra and L. Rifford, Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry, ESAIM Control Optim. Calc. Var., 16 (2010), 695-718.doi: 10.1051/cocv/2009020.

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    D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257.doi: 10.1007/s002220100160.

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    A. Figalli, Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]. Séminaire Bourbaki No. 1009, June 2009.

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    A. Figalli and L. Rifford, Continuity of optimal transport maps and convexity of injectivity domains on small deformations of $\mathbb S^n$, Comm. Pure Appl. Math., 62 (2009), 1670-1706.doi: 10.1002/cpa.20293.

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    A. Figalli, L. Rifford and C. Villani, Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds, Preprint, 2010, available online at http://www.umpa.ens-lyon.fr/~cvillani.

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    A. Figalli, L. Rifford and C. VillaniOn the Ma-Trudinger-Wang curvature on surfaces, To appear in "Calc. Var. Partial Differential Equations,'' available online at http://www.umpa.ens-lyon.fr/~cvillani.

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    A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex, preprint, 2009, will appear in Amer. Math. J., available online at http://www.umpa.ens-lyon.fr/~cvillani.

    [8]

    A. Figalli and C. Villani, Optimal transport and curvature, Notes for a CIME lecture course in Cetraro, June 2008, available online at http://www.umpa.ens-lyon.fr/~cvillani.

    [9]

    J. I. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Trans. Amer. Math. Soc., 353 (2001), 21-40.doi: 10.1090/S0002-9947-00-02564-2.

    [10]

    Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math., 58 (2004), 85-146.doi: 10.1002/cpa.20051.

    [11]

    G. Loeper and C. VillaniRegularity of optimal transport in curved geometry: The nonfocal case, in revision for Duke Math. J., available at http://www.umpa.ens-lyon.fr/~cvillani.

    [12]

    X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), 151-183.doi: 10.1007/s00205-005-0362-9.

    [13]

    C. Villani, "Topics in Optimal Transportation,'' vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.

    [14]

    C. Villani, "Optimal Transport, Old and New,'' Vol. 338 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009.

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