  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, Kim, and McCann, but at the added cost of assuming compact containment
of the supports of both the source and target measures.
