We prove that, for general cost functions on Rn, or for the cost d2/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.
@article{PMIHES_2015__121__81_0, author = {De Philippis, Guido and Figalli, Alessio}, title = {Partial regularity for optimal transport maps}, journal = {Publications Math\'ematiques de l'IH\'ES}, pages = {81--112}, publisher = {Springer Berlin Heidelberg}, address = {Berlin/Heidelberg}, volume = {121}, year = {2015}, doi = {10.1007/s10240-014-0064-7}, language = {en}, url = {https://www.numdam.org/articles/10.1007/s10240-014-0064-7/} }
TY - JOUR AU - De Philippis, Guido AU - Figalli, Alessio TI - Partial regularity for optimal transport maps JO - Publications Mathématiques de l'IHÉS PY - 2015 SP - 81 EP - 112 VL - 121 PB - Springer Berlin Heidelberg PP - Berlin/Heidelberg UR - https://www.numdam.org/articles/10.1007/s10240-014-0064-7/ DO - 10.1007/s10240-014-0064-7 LA - en ID - PMIHES_2015__121__81_0 ER -
%0 Journal Article %A De Philippis, Guido %A Figalli, Alessio %T Partial regularity for optimal transport maps %J Publications Mathématiques de l'IHÉS %D 2015 %P 81-112 %V 121 %I Springer Berlin Heidelberg %C Berlin/Heidelberg %U https://www.numdam.org/articles/10.1007/s10240-014-0064-7/ %R 10.1007/s10240-014-0064-7 %G en %F PMIHES_2015__121__81_0
De Philippis, Guido; Figalli, Alessio. Partial regularity for optimal transport maps. Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 81-112. doi : 10.1007/s10240-014-0064-7. https://www.numdam.org/articles/10.1007/s10240-014-0064-7/
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