# Under what conditions can one find the optimal critic in WGAN?

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.