I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for \begin{equation} \mathbb{E}\min_{w_i \in \left\{-1,1\right\}}|w_1g_1+\cdots+w_ng_n| \text{?} \end{equation}

Basically, I am asking about the minimum expected absolute value of a family of correlated gaussian random variables.

If a good bound can be obtained, what about the same question for more general linear combinations, such as $w_1a_1g_1+\cdots+w_na_ng_n$ in term of $n$ and some norm of $a_i$, say $l_{2}$?