Before revision:  We consider the problem of transporting \mu to \nu in a way that minimizes a cost c: X \times Y \to R of the  form c(x,y)= \Psi(x-y) for some convex function \Psi . For this problem , we find an associated K\"ahler manifold whose orthogonal holomorphic bisectional curvature is proportional to the so-called MTW tensor, which was introduced by Ma, Trudinger, and Wang and plays an essential role in the regularity theory of optimal transport .
After revision:  We consider the problem of transporting \mu to \nu in a way that minimizes a cost c: X \times Y \to R of the form c(x,y)= \Psi(x-y) for some convex function \Psi , we find an associated K\"ahler manifold whose orthogonal holomorphic bisectional curvature is proportional to the so-called MTW tensor, which was introduced by Ma, Trudinger, and Wang and plays an essential role in the regularity theory of optimal transport .