Let $X=(X,d)$ be a (Polish) metric space equipped with the Borel sigma-algebra, and let $\mathcal P(X)$ be the set of probability distributions thereupon. For a function , $f:X \rightarrow \mathbb R$, let $\|f\|_{BL} := \|f\|_\infty + \|f\|_L$, where $\|f\| := \sup_{x \in X}|f(x)|$ and $\|f\|_L := \underset{x,x' \in X,\;x' \ne x}{\sup}\; \frac{|f(x')-f(x)|}{d(x',x)}$.
The bounded-Lipschitz metric (aka Dudley metric) on $\mathcal P(X)$ is defined by
$$ d_{BL}(\mu,\nu) := \sup_{\|f\|_{BL} \le 1}\mathbb E_\mu[f] - \mathbb E_\nu [f]. $$
Now, let $X = \mathbb R^n$ equipped with the euclidean metric. Let $c_1,c_2 \in \mathbb R^n$ and $\Sigma_1,\Sigma_2$ be positive-definite matricies of size $n$.
Question 1. $d_{BL}(\mathcal N(c_1,\Sigma_1),\mathcal N(c_2,\Sigma_2)) = ?$
Question 2. Same question with $\Sigma_2 = \Sigma_1$.
Note that $$ d_{BL}(\mathcal N(c_1,\Sigma),\mathcal N(c_2,\Sigma)) \le W_1(\mathcal N(c_1,\Sigma),\mathcal N(c_2,\Sigma)) = \|c_1-c_2\|_2, $$
where $ W_1(\mu,\nu) := \sup_{\|f\|_L \le 1} \mathbb E_\mu[f] - \mathbb E_\nu[f]$, is the Wasserstein distance between $\mu$ and $\nu$.