American Institute of Mathematical Sciences

May  2011, 30(2): 559-571. doi: 10.3934/dcds.2011.30.559

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

 1 Institut Henri Poincaré & Université Claude Bernard Lyon 1, 11 rue Pierre et Marie Curie 75230 Paris Cedex 05

Received  August 2010 Published  February 2011

In this survey paper I describe the convoluted links between theregularity theory of optimal transport and the geometry of cut locus.
Citation: Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 559-571. doi: 10.3934/dcds.2011.30.559
References:
 [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.  Google Scholar [2] 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.  Google Scholar [3] A. Figalli, Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]. Séminaire Bourbaki No. 1009, June 2009. Google Scholar [4] 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.  Google Scholar [5] 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. Google Scholar [6] A. Figalli, L. Rifford and C. Villani, On the Ma-Trudinger-Wang curvature on surfaces,, To appear in, ().   Google Scholar [7] 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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: The nonfocal case,, in revision for Duke Math. J., ().   Google Scholar [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.  Google Scholar [13] C. Villani, "Topics in Optimal Transportation,'' vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.  Google Scholar [14] C. Villani, "Optimal Transport, Old and New,'' Vol. 338 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. Google Scholar

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References:
 [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.  Google Scholar [2] 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.  Google Scholar [3] A. Figalli, Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]. Séminaire Bourbaki No. 1009, June 2009. Google Scholar [4] 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.  Google Scholar [5] 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. Google Scholar [6] A. Figalli, L. Rifford and C. Villani, On the Ma-Trudinger-Wang curvature on surfaces,, To appear in, ().   Google Scholar [7] 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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [11] G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: The nonfocal case,, in revision for Duke Math. J., ().   Google Scholar [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.  Google Scholar [13] C. Villani, "Topics in Optimal Transportation,'' vol. 58 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.  Google Scholar [14] C. Villani, "Optimal Transport, Old and New,'' Vol. 338 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009. Google Scholar
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