# Continuity and injectivity of optimal maps

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## Abstract

Figalli–Kim–McCann proved in (2009) the continuity and injectivity of optimal maps under the assumption (B3) of nonnegative cross-curvature. In the recent Figalli et al. (Arch Ration Mech Anal 209:747–795, 2013), Figalli et al. (Methods Appl Anal 2013), they extend their results to the assumption (A3w) of Trudinger–Wang (Ann Sc Norm Super Pisa Cl Sci 8:143–174, 2009), and they prove, moreover, the Hölder continuity of these maps. We give here an alternative and independent proof of the extension to (A3w) of the continuity and injectivity of optimal maps based on the sole arguments of Figalli et al. (2009) and on new Alexandrov-type estimates for lower bounds.

## Mathematics Subject Classification

35J96## Notes

### Acknowledgments

The author is very grateful to the Australian National University for its generous hospitality during the period when the majority of this work has been carried out. The author wishes to express his gratitude to Philippe Delanoë, Neil Trudinger, and Xu-Jia Wang for supporting his visit to Canberra, and for several stimulating discussions on optimal transportation. The author is indebted to Neil Trudinger for having introduced him to the problem of regularity of optimal maps. The author wishes also to express his gratitude to Emmanuel Hebey for helpful support and advice during the redaction, and to Jiakun Liu and, again, Neil Trudinger for their valuable comments and suggestions on the manuscript.

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