Local semiconvexity of Kantorovich potentials on non-compact manifolds
Alessio Figalli, Nicola Gigli (2011)
ESAIM: Control, Optimisation and Calculus of Variations
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We prove that any Kantorovich potential for the cost function = /2 on a Riemannian manifold (, ) is locally semiconvex in the “region of interest”, without any compactness assumption on , nor any assumption on its curvature. Such a region of interest is of full -measure as soon as the starting measure does not charge – 1-dimensional rectifiable sets.