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.


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