 Solution

For probability measures $\mu ,\nu$ on ${\mathbb{R}}^d$ with finite second moments, the Knott–Smith optimality criterion states that a transport plan $\pi \in \Pi (\mu ,\nu )$ minimizes the quadratic cost $|x-y{|}^2$ if and only if there is a proper convex function $\Phi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\infty \}$ that is sequentially lower semicontinuous and satisfiesThe subdifferential is characterized bywith $\Phi (x)<\infty$. Thus the transport plan is concentrated on the graph of the subdifferential. No absolute continuity of measures assumption on $\mu$ is needed. Multiplying the cost by $1/2$ leaves the criterion unchanged. The original quadratic optimal mapping result is Knott and Smith, On the optimal mapping of distributions.