Let $\gamma\in\mathcal P_2(\mathbb R^n)$, and equip $\mathcal P_2(\mathbb R^n)$ with the Wasserstein distance $W_2$. Define the relative entropy functional $\operatorname{Ent}_\gamma:\mathcal P_2(\mathbb R^n)\to[0,+\infty]$ by

\begin{align*} \operatorname{Ent}_\gamma(\mu)=\int_{\mathbb R^n}\log\left(\frac{d\mu}{d\gamma}(x)\right)\,d\mu(x) \end{align*}

when $\mu\ll\gamma$, and by $\operatorname{Ent}_\gamma(\mu)=+\infty$ otherwise. In particular $\operatorname{Ent}_\gamma(\gamma)=0$. Define its effective domain by

\begin{align*} \mathcal D(\operatorname{Ent}_\gamma)=\{\mu\in\mathcal P_2(\mathbb R^n):\operatorname{Ent}_\gamma(\mu)<+\infty\}. \end{align*}

Assume that every two measures in $\mathcal P_2(\mathbb R^n)$ can be joined by a constant-speed $W_2$-geodesic. Assume that $\operatorname{Ent}_\gamma$ is strictly displacement convex on $\mathcal D(\operatorname{Ent}_\gamma)$: whenever $\mu_0,\mu_1\in\mathcal D(\operatorname{Ent}_\gamma)$ are distinct and $(\mu_t)_{t\in[0,1]}$ is a constant-speed $W_2$-geodesic joining $\mu_0$ to $\mu_1$, one has

\begin{align*} \operatorname{Ent}_\gamma(\mu_t)<(1-t)\operatorname{Ent}_\gamma(\mu_0)+t\operatorname{Ent}_\gamma(\mu_1) \end{align*}

for every $t\in(0,1)$. Then $\operatorname{Ent}_\gamma$ has at most one minimizer in $\mathcal P_2(\mathbb R^n)$.