  We provide a unifying approach to central limit type theorems for empirical
optimal transport (OT). In general, the limit distributions are characterized
as suprema of Gaussian processes. We explicitly characterize when the limit
distribution is centered normal or degenerates to a Dirac measure. Moreover, in
contrast to recent contributions on distributional limit laws for empirical OT
on Euclidean spaces which require centering around its expectation, the
distributional limits obtained here are centered around the population
quantity, which is well-suited for statistical applications.
  At the heart of our theory is Kantorovich duality representing OT as a
supremum over a function class $\mathcal{F}_{c}$ for an underlying sufficiently
regular cost function $c$. In this regard, OT is considered as a functional
defined on $\ell^{\infty}(\mathcal{F}_{c})$ the Banach space of bounded
functionals from $\mathcal{F}_{c}$ to $\mathbb{R}$ and equipped with uniform
norm. We prove the OT functional to be Hadamard directional differentiable and
conclude distributional convergence via a functional delta method that
necessitates weak convergence of an underlying empirical process in
$\ell^{\infty}(\mathcal{F}_{c})$. The latter can be dealt with empirical
process theory and requires $\mathcal{F}_{c}$ to be a Donsker class. We give
sufficient conditions depending on the dimension of the ground space, the
underlying cost function and the probability measures under consideration to
guarantee the Donsker property. Overall, our approach reveals a noteworthy
trade-off inherent in central limit theorems for empirical OT: Kantorovich
duality requires $\mathcal{F}_{c}$ to be sufficiently rich, while the empirical
processes only converges weakly if $\mathcal{F}_{c}$ is not too complex.
