Projection onto manifold of Gaussian measures by "trunction" of moments

Question

Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of Borel probability measures on $\mathbb{R}^n$ with finite mean and variance; in the sense that $$ \int_{x \in \mathbb{R}^n} \|x\|^p d\mathbb{P}(x) < \infty, $$ for $p=1,2$. Let $\mathcal{G}(\mathbb{R}^n)\subseteq \mathcal{P}(\mathbb{R}^n)$ be the set of Gaussian measures thereon (possibly degenerated).

Does there exist a metric $d$ on $\mathcal{P}_2(\mathbb{R}^n)$ such that the metric projection $$ \inf_{\mathbb{Q}\in \mathcal{G}(\mathbb{R}^n)} d(\mathbb{Q},\mathbb{P}) $$ is given by the Gaussian measure $\hat{P}$ with mean $$ (\mu_{\mathbb{P}})_i=\int_{x \in \mathbb{R}^n} x_i d\mathbb{P}(x) $$ and covariance $$ \left(\Sigma_{\mathbb{P}}\right)_{i,j}\int_{x \in \mathbb{R}^n} \left[ x_i - (\mu_{\mathbb{P}})_i \right] \left[ x_j - (\mu_{\mathbb{P}})_j \right] \mu_{\mathbb{P}}; $$ here $i,j\in \{1,\dots,n\}$?

Answer 1

Such a metric is given by $$d(P,Q):=2\times1\{(\mu_P,\Sigma_P)\ne(\mu_Q,\Sigma_Q)\}+\sup_B|P(B)-Q(B)|$$ for Borel probability measures $P$ and $Q$ on $\mathbb{R}^n$ with finite second moments, where the $\sup$ is taken over all Borel subsets $B$ of $\mathbb{R}^n$.