Hölder regularity of optimal mappings in optimal transportation

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On the local theory of prescribed Jacobian equations (2012, background): “In the optimal transportation case, G(x, y, z) = c(x, y) − z, the c-convexity of sections is proved and used in [5, 15].”
Continuity and injectivity of optimal maps for non-negatively cross-curved costs (2009, methodology): “gauge) invariance to choose coordinates which depend on the point of interest that restore convexity of u(x); our strengthened hypothesis then permits us to exploit Caffarelli’s approach more systematically than Liu was able to do [32].”
Hölder Continuity and Injectivity of Optimal Maps (2011, methodology): “The resulting Alexandrov type estimates enable us to exploit Caffarelli’s approach [5] [7] [8] more systematically than Liu [40] was able to do, to prove continuity and injectivity (see Theorems 8.”
Interior C 2,α Regularity for Potential Functions in Optimal Transportation (2009, methodology): “By making a linear transform of the coordinates, see [12, 19], we may assume that D2 xyc x0 y0 is the identity matrix.”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “ on Y, and denote by Tthe optimal transport map sending fonto g. Finally, suppose that D xc(x;Y) is convex for any x2X. Then u2C1;(X), with = 1=(4n 1) (hence T u2C0;(X)). In a subsequent paper, Liu [83] proved the above result with = 1=(2n 1) and showed that such exponent is optimal. We notice howev”