Consider the following sum:
$$ \inf_{u \in \mathbb{S}^{n-1}} \frac{1}{m} \sum_{i=1}^m \left| \langle a_i, u \rangle \right|, $$ where $\mathbb{S}$ denotes the unit sphere, $$ \mathbb{S}^{n-1} := \left\{ x \in \mathbb{R}^n: \left\| x \right\|_2 = 1 \right\}, $$
and the $a_i$ satisfy $a_i \sim \mathcal{N}(0, I_n)$ individually but are not necessarily independent. (In fact, given a longer vector $g \sim \mathcal{N}(0, I_m)$, every $a_i$ is a shift of $g$ by $i$ positions, cut off at the first $n$ elements. Therefore, the vectors $a_i$ and $a_{i+d}$ are completely independent.)
I'm looking for a lower bound on the quantity
$$ \mathbb{E}\left[\inf_{u \in \mathbb{S}^{n-1}} \frac{1}{m} \sum_{i=1}^m \left| \langle a_i, u \rangle \right| \right]. $$
A trivial lower bound is $0$ (e.g. by "breaking up" the infimum into individual infima) but I think we should be able to do better given the limited dependence in our sample.