Regularity of optimal transport in curved geometry: the nonfocal case

Gregoire Loeper, Cedric Villani

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40 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)431-485
Number of pages55
JournalDuke Mathematical Journal
Issue number3
Publication statusPublished - 2010
Externally publishedYes

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