Maximum of the expectation of maximum of Gaussian variables 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
 A: For the question with the absolute value, the expectation is maximal when the variables are independent (a special case of the Khatri-Sidak inequality).
For the question witout absolute value, it is natural to conjecture that the maximum occurs when the variables form a regular simplex in $L^2$. I think I saw this conjecture formulated once in a paper about stochastic geometry.
Edit: the question appears explicitly here (page 5). It can be reformulated as the question whether the regular simplex maximizes the mean width among simplices inscribed in a Euclidean ball in $\mathbf{R}^{n-1}$. 
A: The case $n=2$ is solved by Charles E. Clark, The Greatest of a Finite Set of Random Variables, Operations Research, Vol. 9, No. 2 (1961), pp. 145-162.
In that case the expected maximum is greatest when the two variables are perfectly negatively correlated. (This makes sense: in this case $\max(X_1,X_2)=\max |X_1|$.)  Clarke does not consider the maximum absolute value and the extension to $n\gt 2$ is not obvious.  It could be that the optimum occurs in different place for even $n$ and odd $n$.
A: I don't want to repeate the answer, so I just point you there.
Your question seems to be the duplicate of this: Maximum of Gaussian Random Variables
If you have trouble understanding the answer, I can clarify if needed.
