The Ma-Trudinger-Wang condition is a curvature condition on a transport cost used to control regularity of optimal transport maps.
For the cost 
on a Riemannian
manifold, its weak form requires
 |
for 
in the open injectivity domain 
and tangent vectors 
with 
.
The cost is smooth at these pairs. For Euclidean squared distance this fourth derivative is zero. The orthogonality
restriction is part of the weak condition.
