The Kantorovich-Rubinstein duality for the optimal transport problem implies that the Wasserstein distance between two distributions $\mu_1$ and $\mu_2$ can be computed as (equation 2 in section 3 in the WGAN paper)
$$W(\mu_1,\mu_2)=\underset{f\in \text{1-Lip.}}{\sup}\left(\mathbb{E}_{x\sim \mu_1}\left[f\left(x\right)\right]-\mathbb{E}_{x \sim \mu_2}\left[f\left(x\right)\right]\right).$$
Under what conditions can one find the optimal $f$ that achieves the maximum? Is it possible to have an analytical expression for $f$ that achieves the maximum in such scenarios?
Any help is deeply appreciated.
