  In this article, we study a class of K\"ahler manifolds defined on tube
domains in $\mathbb{C}^n$, and in particular those which have $O(n) \times
\mathbb{R}^n$ symmetry. For these, we prove a uniqueness result showing that
any such manifold which is complete and has non-negative orthogonal bisectional
curvature ($n \geq 3$) or non-negative bisectional curvature ($n \geq 2$) is
biholomorphically isometric to $\mathbb{C}^n$. We also consider another
curvature tensor called the \emph{orthogonal anti-bisectional curvature}. We
find necessary and sufficient conditions for a complete $O(n)$-symmetric tube
domain to have non-negative orthogonal anti-bisectional curvature and provide
several examples of complete metrics which satisfy this condition. Finally, we
discuss some applications of these spaces within optimal transport. In
particular, we study "synthetic" curvature bounds for non-smooth geometries and
how they can be applied to the rough geometry induced by the Monge cost
$c(x,y)=\|x-y\|$.
