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



Maximum of the expectation of maximum of Gaussian variables
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](http://www.math.kit.edu/stoch/~hug/media/34.pdf) (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 $\mathbf{R}^{n-1}$. 


