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

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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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     title = {Local semiconvexity of {Kantorovich} potentials on non-compact manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {648--653},
     publisher = {EDP-Sciences},
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     year = {2011},
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     zbl = {1228.49047},
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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

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