I'm working on an economic model with a nonlinear production network, and part of the model ends up being described by $\min_i(X_i)$ (the smallest component of $X$), where $X\in\mathbb{R}^n$ is a vector uniformly distributed on the sphere $||X||_2=1$. Is there a known formula for the distribution of $\min_i(X_i)$?
If it's easier, I'd also be interested in any possible solution for alternative $L_p$ norms, e.g. the restriction that $||X||_1=1$.
I'd actually ideally like to be able to calculate some moments of $\min_i(X_i)$, such as $\mathbb{E}[|\min_i(X_i)|^\alpha \mid \min_i(X_i)<0]$ for some $\alpha$, but I suspect that will be a bit too difficult.
An alternative possibility would be to take the limit as $n\rightarrow \infty$ and use extreme value theory.