no. 3
Local semiconvexity of Kantorovich potentials on non-compact manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 648-653

We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n - 1-dimensional rectifiable sets.

DOI : 10.1051/cocv/2010011
Classification : 49Q20, 35J96
Keywords: Kantorovich potential, optimal transport, regularity 
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Figalli, Alessio; Gigli, Nicola. Local semiconvexity of Kantorovich potentials on non-compact manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 648-653. doi: 10.1051/cocv/2010011

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000). | Zbl | MR

[2] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in spaces of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005). | Zbl | MR

[3] D. Cordero-Erasquin, R.J. Mccann and M. Schmuckenschlager, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146 (2001) 219-257. | Zbl | MR

[4] A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. (to appear). | Zbl | MR

[5] A. Figalli, Existence, uniqueness, and regularity of optimal transport maps. SIAM J. Math. Anal. 39 (2007) 126-137. | Zbl | MR

[6] W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | Zbl | MR

[7] N. Gigli, Second order analysis on (𝒫 2 (M),W 2 ) . Memoirs of the AMS (to appear), available at http://cvgmt.sns.it/cgi/get.cgi/papers/gig09/. | Zbl

[8] R.J. Mccann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608. | Zbl | MR

[9] C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin-New York (2009). | Zbl | MR

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