Continuity and injectivity of optimal maps

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 Holder 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.