# On the local geometry of maps with c-convex potentials

@article{Guillen2012OnTL, title={On the local geometry of maps with c-convex potentials}, author={Nestor Guillen and Jun Kitagawa}, journal={Calculus of Variations and Partial Differential Equations}, year={2012}, volume={52}, pages={345-387} }

We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak Ma–Trudinger–Wang condition when the cost is $$C^4$$C4. Moreover, we only require (non-strict) $$c$$c-convexity of the support of the target measure, removing the hypothesis of strong $$c$$c-convexity in a previous result of Figalli et al., but at the added…

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