For a multivariate Gaussian distribution $p(x) = N(x\mid \mu,\Sigma)$, what is $E[\|x-\mu\|]$?
I know from this question that $E[|x-\mu|]=\sigma\sqrt{2/\pi}$ for univariate Gaussians.
But I couldn't find a definition of standard deviation for multivariate Gaussians. Could it be something like $\|\sqrt{\boldsymbol\lambda}\|\sqrt{2/\pi}$, where $\boldsymbol\lambda$ is the eigenvalues of $\Sigma$?