Let $E$ be a polish space, $\mathcal{P}(E)$ the Borel probability measures on $E$ with the topology of weak convergence and $\mathcal{Q} \subset \mathcal{P}(E)$ a convex and compact set.
First, the motivation for my question:
The Kullback Leibler divergence (or relative entropy) $R : \mathcal{P}(E)\times \mathcal{P}(E) \rightarrow [0, \infty]$ is defined by $R(\nu, \mu) = \int \frac{d\nu}{d\mu} \log(\frac{d\nu}{d\mu}) d\mu$ if $\nu$ is absolutely continuous with respect to $\mu$ and $R(\nu, \mu) = \infty$, else.
Let $\theta \in \mathcal{P}(E)$ and $\theta^{\ast}_R := \arg\min_{\mu \in \mathcal{Q}} R(\mu, \theta)$ be the information projection of $\theta$ onto $\mathcal{Q}$. As a consequence of the pythagorean like theorem for relative entropy (c.f. Csiszár (1975), Theorem 2.2) it holds \begin{equation} \label{eq1} \forall \mu \in \mathcal{Q} : R(\theta^{\ast}_R, \theta) + R(\mu, \theta^{\ast}_R) \leq R(\mu, \theta) \end{equation}
My question is whether there exists a metric $d$ on $\mathcal{P}(E)$ which satisfies the same kind of result. So if $\theta^{\ast}_d := \arg\min_{\mu\in\mathcal{Q}} d(\mu, \theta)$, it should hold $$ \forall \mu \in \mathcal{Q}: d(\theta^{\ast}_d, \theta)^2 + d(\mu, \theta^{\ast}_d)^2 \leq d(\mu, \theta)^2 $$
Ideally, the metric $d$ should metrize weak convergence, but I am thankful for any approaches or pointers.
