Let $1 \le p < \infty$, and let $\mu_0,\mu_1 \in \mathcal{P}_p(\mathbb{R}^n)$. Let $\pi \in \Pi(\mu_0,\mu_1)$ be an optimal coupling for the $p$-Wasserstein distance, so that
defines a constant-speed geodesic from $\mu_0$ to $\mu_1$ in $(\mathcal{P}_p(\mathbb{R}^n), W_p)$. Equivalently, $\mu_0$ and $\mu_1$ are the endpoint measures and, for every $s,t \in [0,1]$,