Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

By Alessio Figalli, Ludovic Rifford. 2009

70 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eb9769f45b124bcedd90936df61dc323b45450c6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Partial regularity for optimal transport maps (2012, background): “…now, the MTW condition is known to be satisfied only for very special classes of Riemannian manifolds, such as spheres, their products, their quotients and submersions, and their perturbations [10, 11, 20, 22, 25, 29, 32], and for instance it is known to fail on sufficiently flat ellipsoids [23].”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, methodology): “• By Proposition C.6, whose proof was communicated to us by Rifford, I(M) is semiconvex (in a sense which is made precise in Appendix C.3).”
NECESSARY AND SUFFICIENT CONDITIONS FOR CONTINUITY OF OPTIMAL TRANSPORT MAPS ON RIEMANNIAN MANIFOLDS (2011, background): “In particular, exp−1y (S(y)) ⊂ I(y) is always a convex set.”
Continuity, curvature, and the general covariance of optimal transportation (2007, background): “Delanoë & Ge have quantified this observation away from the cut locus on the round sphere [15], while the subtleties associated with the cut locus have been resolved on S2 by Figalli & Rifford [19]; see also Loeper & Villani [33].”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “dition of nonnegativity of sectional curvatures. In conclusion, except for some special cases, the optimal map is nonsmooth. Some examples of manifolds satisfying the MTW condition have been found in [87, 88, 76, 77, 56, 58]: - Rn and Tn satisfy MTW(0). - Sn, its quotients (like RP n), and its subme”