Let $n \in \mathbb N$. Let $\mathcal P_{2}^{\infty}(\mathbb R^n)$ denote the class of strictly positive $C^\infty$ probability densities $\rho:\mathbb R^n\to(0,\infty)$ such that $\int_{\mathbb R^n}\rho\,d\mathcal L^n=1$, $\int_{\mathbb R^n}|x|^2\rho(x)\,d\mathcal L^n(x)<\infty$, and the decay assumptions needed for the formal Wasserstein first- and second-variation formulas hold. Let

\begin{align*} \mathcal F:\mathcal P_{2}^{\infty}(\mathbb R^n)\to\mathbb R \end{align*}

be twice differentiable in the formal $W_2$ calculus. Assume that, for every $\rho\in\mathcal P_{2}^{\infty}(\mathbb R^n)$ and every $\phi\in C_c^\infty(\mathbb R^n)$,

\begin{align*} \operatorname{Hess}_{W_2}\mathcal F(\rho)[\nabla\phi,\nabla\phi]\ge 0. \end{align*}

Then $\mathcal F$ is formally displacement convex along compactly supported smooth Wasserstein geodesics in the following sense: if $I\subset\mathbb R$ is an interval and $(\rho_t)_{t\in I}\subset\mathcal P_{2}^{\infty}(\mathbb R^n)$ is a smooth formal $W_2$ geodesic for which there exists a smooth family $(\phi_t)_{t\in I}\subset C_c^\infty(\mathbb R^n)$ satisfying

\begin{align*} \partial_t\rho_t+\operatorname{div}(\rho_t\nabla\phi_t)=0 \end{align*}

and whose velocity field is $\nabla\phi_t$ at time $t$, then the function

\begin{align*} I\to\mathbb R,\qquad t\mapsto \mathcal F[\rho_t] \end{align*}

is convex on $I$.