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 $i,j\in \{1,\dots,n\}$?



Projection onto manifold of Gaussian measures by "trunction" of moments
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 $\mathbb{R}^n$.


