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

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## Abstract

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 \(C^4\). Moreover, we only require (non-strict) \(c\)-convexity of the support of the target measure, removing the hypothesis of strong \(c\)-convexity in a previous result of Figalli et al., but at the added cost of assuming compact containment of the supports of both the source and target measures.

## Keywords

Optimal transport c-convex geometry Monge-Ampère equation Mass transport Regularity theory Degenerate elliptic equations## Mathematics Subject Classification

35J96 90C08 49N60 52A41## References

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