Let X and Y be domains of R^n equipped with  probability measures \mu and \nu , respectively . 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 . We also show that relative c-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete K\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to \mathbb{C}^2. We also address a question in mathematical finance raised by Pal and Wong on the regularity of \textit{pseudo-arbitrages}, or investment strategies which outperform the market.
Let X and Y be domains of R^n equipped with respective probability measures \mu and \nu . We consider the problem of optimal transport from \mu to \nu with respect to a cost function c: X \times Y \to R . To ensure that the solution to this problem is smooth, it is necessary to make several assumptions about the structure of the domains and the cost function. In particular, Ma, Trudinger, and Wang established regularity estimates when the domains are stronglyrelatively c-convex with respect to each other and cost function has non-negativeMTW tensor. For cost functions 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 MTW tensor . We also show that relative c-convexity geometrically corresponds to geodesic convexity with respect to a dual affine connection. Taken together, these results provide a geometric framework for optimal transport which is complementary to the pseudo-Riemannian theory of Kim and McCann. We provide several applications of this work. In particular, we find a complete K\"ahler surface with non-negative orthogonal bisectional curvature that is not a Hermitian symmetric space or biholomorphic to \mathbb{C}^2. We also address a question in mathematical finance raised by Pal and Wong on the regularity of \textit{pseudo-arbitrages}, or investment strategies which outperform the market.
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