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 the regularity 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 - A, 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.  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.  doi: 10.1007/s002220100160.  Google Scholar

[3]

A. Figalli, Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]., S\'eminaire Bourbaki No. 1009, (1009).   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.  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).   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).   Google Scholar

[8]

A. Figalli and C. Villani, Optimal transport and curvature,, Notes for a CIME lecture course in Cetraro, (2008).   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.  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.  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.  doi: 10.1007/s00205-005-0362-9.  Google Scholar

[13]

C. Villani, "Topics in Optimal Transportation,'', vol. \textbf{58} of Graduate Studies in Mathematics, 58 (2003).   Google Scholar

[14]

C. Villani, "Optimal Transport, Old and New,'', Vol. \textbf{338} of Grundlehren der mathematischen Wissenschaften, 338 (2009).   Google Scholar

show all references

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.  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.  doi: 10.1007/s002220100160.  Google Scholar

[3]

A. Figalli, Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]., S\'eminaire Bourbaki No. 1009, (1009).   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.  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).   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).   Google Scholar

[8]

A. Figalli and C. Villani, Optimal transport and curvature,, Notes for a CIME lecture course in Cetraro, (2008).   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.  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.  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.  doi: 10.1007/s00205-005-0362-9.  Google Scholar

[13]

C. Villani, "Topics in Optimal Transportation,'', vol. \textbf{58} of Graduate Studies in Mathematics, 58 (2003).   Google Scholar

[14]

C. Villani, "Optimal Transport, Old and New,'', Vol. \textbf{338} of Grundlehren der mathematischen Wissenschaften, 338 (2009).   Google Scholar

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