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.

An energy has $\lambda$-displacement convexity when the displayed inequality holds along every displacement interpolation between endpoints in its domain. The case $\lambda =0$ is displacement convexity; $\lambda >0$ gives uniform convexity along these metric geodesics. This is distinct from convexity along ordinary linear mixtures of probability measures. On subintervals, the inequality is equivalent to convexity of $t\mapsto E({\mu }_t)-\lambda W_2({\mu }_0,{\mu }_1{)}^2{t}^2/2$; if the energy is twice differentiable, it is equivalent to the corresponding second derivative lower bound. Without that regularity the lower bound is interpreted using distributions.