Let $n \in \mathbb N$, let $\mu,\nu \in \mathcal P_2(\mathbb R^n)$ be Borel probability measures with finite second moments, and let $\Pi(\mu,\nu)$ denote the set of Borel probability measures $\pi$ on $\mathbb R^n \times \mathbb R^n$ whose first marginal is $\mu$ and whose second marginal is $\nu$. Assume the standard [Kantorovich duality](/theorems/6799) theorem for the quadratic cost on $\mathbb R^n$ under finite second-moment hypotheses in the following form: for every quadratic-cost optimal plan $\pi_0 \in \Pi(\mu,\nu)$, there exist Borel functions $f,g: \mathbb R^n \to \mathbb R$ that are integrable with respect to $\mu$ and $\nu$, respectively, satisfy $f(x)+g(y)\le |x-y|^2$ for all $x,y\in\mathbb R^n$, attain the dual value, and satisfy [complementary slackness](/theorems/2559) $f(x)+g(y)=|x-y|^2$ for $\pi_0$-almost every $(x,y)$. Then

\begin{align*} \inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb R^n \times \mathbb R^n} |x-y|^2 \, d\pi(x,y) = \int_{\mathbb R^n} |x|^2 \, d\mu(x) + \int_{\mathbb R^n} |y|^2 \, d\nu(y) - 2 \sup_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb R^n \times \mathbb R^n} x \cdot y \, d\pi(x,y). \end{align*}

Moreover, there exists at least one minimizer for the left-hand side. If $\pi_0 \in \Pi(\mu,\nu)$ attains this infimum, then there exists a proper lower semicontinuous convex function $\varphi: \mathbb R^n \to (-\infty,\infty]$ such that its convex subdifferential graph

\begin{align*} \operatorname{Graph}(\partial \varphi):=\{(x,y) \in \mathbb R^n \times \mathbb R^n : y \in \partial \varphi(x)\} \end{align*}

is a Borel subset of $\mathbb R^n \times \mathbb R^n$ and

\begin{align*} \pi_0(\operatorname{Graph}(\partial \varphi))=1. \end{align*}