Let $\mu_0,\mu_1 \in \mathcal{P}_2(\mathbb{R}^n)$, where $\mathcal{P}_2(\mathbb{R}^n)$ denotes the Borel probability measures on $\mathbb{R}^n$ with finite second moment. Suppose there exists a Borel map $T:\mathbb{R}^n \to \mathbb{R}^n$ such that
is an optimal transport plan from $\mu_0$ to $\mu_1$ for the quadratic cost $c:\mathbb{R}^n\times\mathbb{R}^n\to[0,\infty)$, $c(x,y)=|x-y|^2$. For each $t \in [0,1]$, define the Borel map $F_t:\mathbb{R}^n \to \mathbb{R}^n$ by
Then $(\mu_t)_{t\in[0,1]}$ is a constant-speed $W_2$ geodesic from $\mu_0$ to $\mu_1$; equivalently, $\mu_0$ and $\mu_1$ are the endpoints and, for all $s,t \in [0,1]$,