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Local semiconvexity of Kantorovich potentials on non-compact manifolds*

Published online by Cambridge University Press:  31 March 2010

Alessio Figalli  and
Nicola Gigli
Alessio Figalli
Affiliation:
Centre de Mathématiques Laurent Schwartz, UMR 7640, École Polytechnique, 91128 Palaiseau, France. figalli@math.polytechnique.fr
Nicola Gigli
Affiliation:
University of Bordeaux, France. nicolagigli@googlemail.com
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Abstract

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.

Keywords

Kantorovich potentialoptimal transportregularity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 648 - 653
Copyright
© EDP Sciences, SMAI, 2010

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