Let $Z_n \in \mathbb{R}^n$, $Z_n \sim N(0, I_n)$ be a gaussian random vector, where $I_n$ is the identity matrix. The unit ball is defined as
$$ L_1 = \left[X \in \mathbb{R}^n: \| X \|_1 \leq 1 \right]$$
The orthogonal projection onto $L_1$ is
$$ P_{L_1}(Z_n) = \underset{\mathbf{x} \in L_1}{\operatorname{argmin}} \| \mathbf{x} - Z_n \|_2 $$
I need to find a nontrivial (as a function of the dimension $n$) lower bound for
$$ \mathbb{P} \left( \max_{i \in (1,\dots,n)} [P_{L_1}(Z_n)]_i = 0 \right), $$
where $ [P_{L_1}(Z_n)]i $ is the $i$-th coordinate of $P_{L_1}(Z_n)$. Does anyone have an idea (or references) on how to solve this problem? In particular, is there a general expression for that (or integral form)?
