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/

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