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We address the problem of proving a Central Limit Theorem for the empiricaloptimal transport cost, $\sqrt{n}\{\mathcal{T}_c(P_n,Q)-\mathcal{W}_c(P,Q)\}$,in the semi discrete case, i.e when the distribution $P$ is finitely supported.We show that the asymptotic distribution is the supremun of a centered Gaussianprocess which is Gaussian under some additional conditions on the probability$Q$ and on the cost. Such results imply the central limit theorem for the$p$-Wassertein distance, for $p\geq 1$. Finally, the semidiscrete frameworkprovides a control on the second derivative of the dual formulation, whichyields the first central limit theorem for the optimal transport potentials.

A Central Limit Theorem for Semidiscrete Wasserstein Distances