Local semiconvexity of Kantorovich potentials on non-compact manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 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 : https://doi.org/10.1051/cocv/2010011
Classification:  49Q20,  35J96
Keywords: Kantorovich potential, optimal transport, regularity
@article{COCV_2011__17_3_648_0,
author = {Figalli, Alessio and Gigli, Nicola},
title = {Local semiconvexity of Kantorovich potentials on non-compact manifolds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {648-653},
doi = {10.1051/cocv/2010011},
zbl = {1228.49047},
mrnumber = {2826973},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_3_648_0}
}

Figalli, Alessio; Gigli, Nicola. Local semiconvexity of Kantorovich potentials on non-compact manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 648-653. doi : 10.1051/cocv/2010011. http://www.numdam.org/item/COCV_2011__17_3_648_0/

[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). | MR 1857292 | Zbl 0957.49001

[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). | MR 2129498 | Zbl 1090.35002

[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. | MR 1865396 | Zbl 1026.58018

[4] A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. (to appear). | MR 2607536 | Zbl 1198.49044

[5] A. Figalli, Existence, uniqueness, and regularity of optimal transport maps. SIAM J. Math. Anal. 39 (2007) 126-137. | MR 2318378 | Zbl 1132.28322

[6] W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113-161. | MR 1440931 | Zbl 0887.49017

[7] N. Gigli, Second order analysis on $\left({𝒫}_{2}\left(M\right),{W}_{2}\right)$ . Memoirs of the AMS (to appear), available at http://cvgmt.sns.it/cgi/get.cgi/papers/gig09/. | Zbl 1253.58008

[8] R.J. Mccann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. 11 (2001) 589-608. | MR 1844080 | Zbl 1011.58009

[9] C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin-New York (2009). | MR 2459454 | Zbl 1156.53003