Given a normal vector $X$ such that $\mathbb{E}(X)=0$ and $Cov(X)=Id$, is it possible to get an expression for $$\mathbb{E}(\|A X\|_1)$$ where $A$ is a given matrix. I know that in dimension 1, we have $$\mathbb{E}(|a X|) = |a|\sqrt{\frac{2}{\pi}}$$ but can we generalize this result? Thanks.
Charles