Regularity of optimal transport in curved geometry: the nonfocal case

Gregoire Loeper, Cedric Villani

Research output: Contribution to journalArticleResearchpeer-review

35 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)431-485
Number of pages55
JournalDuke Mathematical Journal
Volume151
Issue number3
DOIs
Publication statusPublished - 2010
Externally publishedYes

Cite this

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Regularity of optimal transport in curved geometry: the nonfocal case. / Loeper, Gregoire; Villani, Cedric.

In: Duke Mathematical Journal, Vol. 151, No. 3, 2010, p. 431-485.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Loeper, Gregoire

AU - Villani, Cedric

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AB - 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.

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DO - 10.1215/00127094-2010-003

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JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

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