The Support of an Optimal Coupling is Cyclically Monotone

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$ and 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 and let $\pi\in\Pi(\mu,\nu)$ be an optimal coupling of $\mu$ and $\nu$. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii) the measure $\pi$ is a Borel measure on the metric space $(\mathbb{R}^{d+d},d_{E})$ of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (ii), so its support $\operatorname{supp}\pi$ is defined.

1. (Cyclical monotonicity of the support)(i) The set $\operatorname{supp}\pi$ is cyclically monotone.