Duke Mathematical Journal

Regularity of optimal transport in curved geometry: The nonfocal case

Grégoire Loeper and Cédric Villani

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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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Duke Math. J. Volume 151, Number 3 (2010), 431-485.

First available in Project Euclid: 8 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 49Q20: Variational problems in a geometric measure-theoretic setting


Loeper, Grégoire; Villani, Cédric. Regularity of optimal transport in curved geometry: The nonfocal case. Duke Math. J. 151 (2010), no. 3, 431--485. doi:10.1215/00127094-2010-003. https://projecteuclid.org/euclid.dmj/1265637659

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