  We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$,
$d\ge 1$, with a convex cost function given by the cumulant generating function
of a probability measure. Examples include the Wasserstein-2 transport whose
cost function is the square of the Euclidean distance and corresponds to the
cumulant generating function of the multivariate standard normal distribution.
The optimal coupling is usually described via an extended notion of
convex/concave functions and their gradient maps. These extended notions are
nonintuitive and do not satisfy useful inequalities such as Jensen's
inequality. Under mild regularity conditions, we show that all such extended
gradient maps can be recovered as the usual supergradients of a nonnegative
concave function on the space of probability distributions. This embedding
provides a universal geometry for all such optimal transports and an unexpected
connection with information geometry of exponential families of distributions.
