Calculus of Variations and Geometric Measure Theory

A. Figalli - N. Gigli

Local semiconvexity of Kantorovich potentials on non-compact manifolds

created by figalli on 09 Oct 2009
modified by gigli on 13 Oct 2009


Accepted Paper

Inserted: 9 oct 2009
Last Updated: 13 oct 2009

Journal: ESAIM Control Optim. Calc. Var.
Year: 2009


We prove that any Kantorovich potential for the cost function $c=d^2/2$ 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 $\mu$-measure as soon as the starting measure $\mu$ does not charge $n-1$-dimensional rectifiable sets.