Abstract
| Original language | English |
|---|---|
| Pages (from-to) | 431-485 |
| Number of pages | 55 |
| Journal | Duke Mathematical Journal |
| Volume | 151 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2010 |
| Externally published | Yes |
Cite this
}
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 journal › Article › Research › peer-review
TY - JOUR
T1 - Regularity of optimal transport in curved geometry: the nonfocal case
AU - Loeper, Gregoire
AU - Villani, Cedric
PY - 2010
Y1 - 2010
N2 - 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.
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.
UR - http://projecteuclid.org/download/pdf_1/euclid.dmj/1265637659
U2 - 10.1215/00127094-2010-003
DO - 10.1215/00127094-2010-003
M3 - Article
VL - 151
SP - 431
EP - 485
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
SN - 0012-7094
IS - 3
ER -