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

Abstract / Introduction Related Papers Cited by
  • 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.


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    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.


    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.


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


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


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


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

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