Abstract
We explore some geometric and analytic consequences of a curvature condition introduced by Ma, Trudinger, and Wang in relation to the smoothness of optimal transport in curved geometry. We discuss a conjecture according to which a strict version of the Ma-Trudinger-Wang condition is sufficient to prove regularity of optimal transport on a Riemannian manifold. We prove this conjecture under a somewhat restrictive additional assumption of nonfocality; at the same time, we establish the striking geometric property that the tangent cut locus is the boundary of a convex set. Partial extensions are presented to the case when there is no pure focalization on the tangent cut locus
Citation
Grégoire Loeper. Cédric Villani. "Regularity of optimal transport in curved geometry: The nonfocal case." Duke Math. J. 151 (3) 431 - 485, 15 February 2010. https://doi.org/10.1215/00127094-2010-003
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