Let $(\mathcal{X}, d)$ be a Polish space, $p \geq 1$, $\mathcal{P}_p(\mathcal{X})$ be the space of probability measures over $\mathcal{X}$ such that the $p$-Wasserstein metric between any two elements in $\mathcal{P}_p(\mathcal{X})$ is finite, and $\mathcal{P}_{\theta}$ is a subspace of $\mathcal{P}_p(\mathcal{X})$. For any $\mu \in \mathcal{P}_p(\mathcal{X})$, does it guarantee that the projection $arg \min_{\nu \in \mathcal{P}_{\theta}} W_p(\mu, \nu)$ has a unique solution?
Thank you.