Given these two relationships, what is true about them?
$$ (i): \min_{|| e ||_2 \leq c} |p^H x - e^H x |^2 \geq or \leq | \min_{|| e ||_2 \leq c} p^H x - e^H x |^2 $$
$$ (ii): \max_{|| e ||_2 \leq c} |p^H x - e^H x |^2 \geq or \leq | \max_{|| e ||_2 \leq c} p^H x - e^H x |^2 $$
where $|| e ||_2 \leq || p ||_2 $, $ e $ is a complex vector and the vectors $ p $, $ x $ are known. Also $ c $ is known.
I have found that under the condition provided $|| e ||_2 \leq || p ||_2 $, then $(i)$ and $(ii)$ hold with equality which to me is somewhat counter-intuitive.
The solution I got for (i) is $ e = \frac{p}{|| p ||_2} c $ and for (ii) is $ e = -\frac{p}{|| p ||_2} c $.
Can anyone explain why the order of the min operator does not matter? Thank you in advance.