Let $A\in\mathbb{R}^{n\times n}$ invertible, and $b:=(0, \alpha, 2\alpha, \ldots, (n-1)\alpha)\in\mathbb{R}^n$ where $\alpha>0$ is such that $\|b\|_2=1$.
Are you aware of a way to estimate the minimum $$ m_b :=\min_{\sigma\in S_n}\|AP_\sigma\cdot b\| $$
by anything better than just $m_b \leq \|A\|$?
Here, $S_n$ is the symmetric group on $n$ letters, and $P_\sigma\in\mathbb{R}^{n\times n}$ is the permutation matrix associated to $\sigma\in S_n$.
(Context: The minimum $m_b$ arises as a multiplicative factor that relates an input quantity to an associated output, where the goal is for the output to be as small as possible given a fixed input.)