Let $E$ be a polish and let $P$ and $Q$ be finitely supported probability measures on $X$. What conditions are required to ensure that: for every $\delta>0$ there exists a $\delta$-optimal transport plan $T:\operatorname{supp}(P)\rightarrow \operatorname{supp}(Q)$ i.e. $$ \int \, \|T(x)\,-x\|^p\,P(dx) \le \mathcal{W}_p^p(P,Q) + \delta? $$
Where here $p\in [1,\infty)$.
