Local semiconvexity of Kantorovich potentials on non-compact manifolds*

Alessio Figalli; Nicola Gigli

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 3, page 648-653
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Figalli, 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

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  1. 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).  
  2. 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).  
  3. 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.  
  4. A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds. Israel J. Math. (to appear).  
  5. A. Figalli, Existence, uniqueness, and regularity of optimal transport maps. SIAM J. Math. Anal.39 (2007) 126–137.  
  6. W. Gangbo and R.J. McCann, The geometry of optimal transportation. Acta Math.177 (1996) 113–161.  
  7. N. Gigli, Second order analysis on ( 𝒫 2 ( M ) , W 2 ) . Memoirs of the AMS (to appear), available at .  URIhttp://cvgmt.sns.it/cgi/get.cgi/papers/gig09/
  8. R.J. McCann, Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal.11 (2001) 589–608.  
  9. C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer-Verlag, Berlin-New York (2009).  

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