Stability results on the smoothness of optimal transport maps with general costs

We prove some stability results concerning the smoothness of optimal transport maps with general cost functions. In particular, we show that the smoothness of optimal transport maps is an open condition with respect to the cost function and the densities. As a consequence, we obtain regularity for a large class of transport problems, where the cost does not necessarily satisfy the MTW condition. (C) 2016 Published by Elsevier Masson SAS.