If z is a centered gaussian random variable and $ x_1 ,x_2 ,..,x_n ,y_1,y_2,..,y_n $ are points in $ \mathbb{R}^{2n} $ satisfying $ |x_i-x_j |_2 \leq |y_i -y_j |_2 \ \ \ \forall i,j \in [n] $ then $ E \ max_{i=1} ^n <x_i, z> \ \leq \ E \ max_{i=1} ^n <y_i, z> . $
Apparently this is not true for other distributions even for centered ones. The proofs I've found are very long and confusing, does anyone have any intuitive explanation for why the gaussian distribution works? Is it purely coincidence that the gaussian distribution also shows up in the central limit theorem?
A good intuition would be to consider the shape of the level sets of the likelihood. That is: the level sets of the density of a Gaussian correspond exactly to the level sets of the $\ell_2$ norm. The $\ell_2$ norm is induced by an inner product, so generally all your linear algebra and geometry intuitions usually transfer pretty well into the Gaussian case. For example, maximum likelihood estimates can reduce to least squares, since the log-likelihood is the $\ell_2$ norm.
Other centered distributions might be transformed in very weird ways when mapped through an inner product, but jointly Gaussian random variables behave nicely through linear transformations.
Hope that helps!