Maximum of absolute value of linear combinations with i.i.d random variables Suppose $x_{1},\dots,x_{n}$ are i.i.d random variables with density
$p(x_{i})=exp(-|x_{i}|)/2$. Denote column vector $x=(x_{1},\dots,x_{n})^{T}$
Let $C\in\mathbb{R}^{n\times n}$ be a matrix with unit euclidean norm rows
$c_{i}^T$ representing linear combinations. 
Define $w(C)=\mathbb{E}_{x}\{\max_{j\in[n]}|c_{j}^Tx|\}$
Which $C$ maximizes $w(C)$?
 A: This does not fully answer your question, but it might be helpful. You can write $\mathbf{x}=r \mathbf{u}$ such that $r$ is an exponentially distributed random variable with mean $\frac{\sqrt{n}}{(n-1)!}$  and $\mathbf{u}$ is a vector uniformly sampled from the surface of the unit $\ell_1$ ball. With independece of $r$ and $\mathbf{u}$, and using the fact that linear (and in general convex) functions attain their maximum on the boundary of their domain, you can reduce your problem in finding $C$ that maximizes $\mathbb{E}_\mathbf{u}\left[\sup_{\mathbf{c}\in\mathcal{P}}\mathbf{c}^\mathrm{T}\mathbf{u}\right]$, where $\mathcal{P}$ is the convex hull of $\lbrace\mathbf{c}_i\rbrace_{i=1}^n$. So, basically you want to find a polytope with $n$ vertices inscribed in the unit Euclidean ball such that it has maximum width in the sense defined above. I'm not sure if you can solve this exactly, but using symmetries of $\ell_1$ ball you might be able to find some properties of $\mathcal{P}$. For instance, I guess $\mathcal{P}$ would be invariant under permutation.
Correction: Since $r=\Vert \mathbf{x}\Vert_1$ is the sum of $n$ iid $\text{Exp(1)}$ random variables, it should have $\text{Erlang}(n,1)$ distribution itself rather than exponential.
