Let $(X,d)$ be a complete separable [metric space](/page/Metric%20Space) such that every pair of points in $X$ is joined by a constant-speed geodesic. Let $\operatorname{Geo}(X)$ denote the set of all constant-speed geodesics $\gamma:[0,1]\to X$, equipped with the Borel $\sigma$-algebra inherited from $C([0,1];X)$ with the uniform metric. For each $t\in[0,1]$, let $e_t:\operatorname{Geo}(X)\to X$ be the Borel evaluation map $e_t(\gamma)=\gamma(t)$.

Let $1\le p<\infty$, let $\mu_0,\mu_1\in\mathcal P_p(X)$, and let $\pi\in\Pi(\mu_0,\mu_1)$ be an optimal coupling for the $p$-Wasserstein distance, meaning

\begin{align*} \int_{X\times X} d(x,y)^p\,d\pi(x,y)=W_p(\mu_0,\mu_1)^p. \end{align*}

Suppose that $\Pi$ is a probability measure on $\operatorname{Geo}(X)$ satisfying

\begin{align*} (e_0,e_1)_\#\Pi=\pi. \end{align*}

For each $t\in[0,1]$, define

\begin{align*} \mu_t:=(e_t)_\#\Pi. \end{align*}

Then $\mu_t\in\mathcal P_p(X)$ for every $t\in[0,1]$, and the curve $[0,1]\ni t\mapsto\mu_t\in\mathcal P_p(X)$ is a constant-speed geodesic from $\mu_0$ to $\mu_1$:

\begin{align*} W_p(\mu_s,\mu_t)=|t-s|\,W_p(\mu_0,\mu_1) \end{align*}

for all $s,t\in[0,1]$.