Local semiconvexity of Kantorovich potentials on non-compact manifolds*
ESAIM: Control, Optimisation and Calculus of Variations (2011)
- Volume: 17, Issue: 3, page 648-653
- ISSN: 1292-8119
Access Full Article
top
Abstract
topHow to cite
topFigalli, Alessio, and Gigli, Nicola. "Local semiconvexity of Kantorovich potentials on non-compact manifolds*." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 648-653. <http://eudml.org/doc/221919>.
@article{Figalli2011,
abstract = {
We prove that any Kantorovich potential for the cost function
c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex
in the “region of interest”, without any compactness assumption
on M, nor any assumption on its curvature. Such a region of
interest is of full μ-measure as soon as the starting measure
μ does not charge n – 1-dimensional rectifiable sets.
},
author = {Figalli, Alessio, Gigli, Nicola},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Kantorovich potential; optimal transport; regularity},
language = {eng},
month = {8},
number = {3},
pages = {648-653},
publisher = {EDP Sciences},
title = {Local semiconvexity of Kantorovich potentials on non-compact manifolds*},
url = {http://eudml.org/doc/221919},
volume = {17},
year = {2011},
}
TY - JOUR
AU - Figalli, Alessio
AU - Gigli, Nicola
TI - Local semiconvexity of Kantorovich potentials on non-compact manifolds*
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/8//
PB - EDP Sciences
VL - 17
IS - 3
SP - 648
EP - 653
AB -
We prove that any Kantorovich potential for the cost function
c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex
in the “region of interest”, without any compactness assumption
on M, nor any assumption on its curvature. Such a region of
interest is of full μ-measure as soon as the starting measure
μ does not charge n – 1-dimensional rectifiable sets.
LA - eng
KW - Kantorovich potential; optimal transport; regularity
UR - http://eudml.org/doc/221919
ER -
References
top- L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000).
- L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in spaces of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005).
- D. Cordero-Erasquin, R.J. McCann and M. Schmuckenschlager, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math.146 (2001) 219–257.
- A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. (to appear).
- A. Figalli, Existence, uniqueness, and regularity of optimal transport maps. SIAM J. Math. Anal.39 (2007) 126–137.
- W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161.
- N. Gigli, Second order analysis on . Memoirs of the AMS (to appear), available at . URIhttp://cvgmt.sns.it/cgi/get.cgi/papers/gig09/
- R.J. McCann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal.11 (2001) 589–608.
- C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin-New York (2009).
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.