Optimal Coupling of Two Probability Measures with Finite Second Moment

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment, let $\Pi(\mu,\nu)$ be the set of their couplings, $I(\pi)$ the quadratic cost of $\pi\in\Pi(\mu,\nu)$, and $W_{2}(\mu,\nu)$ their quadratic Wasserstein distance.

(Optimal coupling)¶ A coupling $\pi\in\Pi(\mu,\nu)$ is optimal if