Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of $$\mathbb{E}\max_{1\le i\le n}|X_i|$$ and $$\mathbb{E}\max_{1\le i\le n}X_i$$ among all correlation structures of $X$.

Many thanks!

John