Displacement convexity

Remarks

Example: let $V \colon \R^n \to \R$ be convex and $F(\mu) = \int V \, d\mu$. Then $F(\mu_t) = \int V\big((1 - t) x + t \nabla \varphi(x)\big) \, d\mu_0(x)$, and the integrand is convex in $t$ for each $x$, so $F$ is displacement convex.

Non-example: on $\R$, let $V(x) = -x^2$, let $\mu_0$ be uniform on $[0, 1]$ and $\mu_1$ uniform on $[2, 3]$. The optimal map is $x \mapsto x + 2$, so $\mu_t$ is uniform on $[2t, 2t + 1]$ and $F(\mu_t) = -\int_0^1 (x + 2t)^2 \, dx = -\left(\tfrac13 + 2t + 4t^2\right)$, which is strictly concave in $t$.

The interpolation $\mu_t$ is a constant-speed geodesic for the Wasserstein distance $W_2$, so displacement convexity is convexity along $W_2$-geodesics. The functional $F$ is $\lambda$-displacement convex, for a real $\lambda$, if $t \mapsto F(\mu_t) - \frac{\lambda}{2} t^2 W_2(\mu_0, \mu_1)^2$ is convex. McCann showed that the entropy $\int \rho \log \rho \, dx$ of a density $\rho$ is displacement convex on $\R^n$. The interaction energy $\iint W(x - y) \, d\mu(x) \, d\mu(y)$ with $W$ convex is also displacement convex. It need not be convex in the usual sense: for $W(z) = \abs{z}^2$ it equals twice the variance of $\mu$, which is concave under mixtures $(1 - t)\mu_0 + t\mu_1$.