  Let $X$ and $Y$ be domains of $\mathbb{R}^n$ 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 $c: X \times Y
\to \mathbb{R}$. 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 strongly \textit{relatively
$c$-convex} with respect to each other and cost function has non-negative
\textit{MTW 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 anti-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.
