Let $n,N\in\mathbb N$. Let $\Omega\subset\mathbb R^n$ be compact, let $\rho_0\in L^1(\Omega,\mathcal B(\Omega),\mathcal L^n\!\restriction_\Omega)$ satisfy $\rho_0\ge 0$ $\mathcal L^n$-a.e. on $\Omega$, and define the finite Borel measure $\mu_0$ on $\Omega$ by
for every $A\in\mathcal B(\Omega)$. Assume $\mu_0(\Omega)>0$. Let $y_1,\dots,y_N\in\mathbb R^n$ be distinct points, and let $m_1,\dots,m_N\in(0,\infty)$ satisfy
Moreover, $T_w$ is an optimal transport map from $\mu_0$ to $\sum_{i=1}^N m_i\delta_{y_i}$ for the quadratic cost $c:\Omega\times\{y_1,\dots,y_N\}\to[0,\infty)$ defined by