We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C1,1 estimates for the potential and, second, interior Hölder estimates for second derivatives. In particular, we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary.