Stability of a 4th-order curvature condition arising in optimal transport theory

A certain curvature condition, introduced by Ma, Trudinger and Wang in relation with the regularity of optimal transport, is shown to be stable under Gromov–Hausdorff limits, even though the condition implicitly involves fourth order derivatives of the Riemannian metric. Two lines of reasoning are presented with slightly different assumptions, one purely geometric, and another one combining geometry and probability. Then a converse problem is studied: prove some partial regularity for the optimal transport on a perturbation of a Riemannian manifold satisfying a strong form of the Ma–Trudinger–Wang condition.