Displacement interpolation (${\mu }_t=((1-t)\mathrm{Id}+tT{)}_{\#}\mu$)

In Euclidean space, push an optimal transport plan $\pi$ forward by $(x,y)\mapsto (1-t)x+ty$. The resulting probability measures form a constant speed curve for the p-Wasserstein distance:If the plan is induced by a transport map $T$, this becomes the displayed title formula.