I am trying to prove or provide a counter-example for the following statement:
If $x$ and $y$ are two complex vectors in $\mathbb{C}^n$ that are perpendicular to each other, i.e., $x^{H} y = 0$ where $H$ denotes Hermitian, then
$\|x\|_{\infty} \leq \|x + y \|_{\infty}$.
This is very easy to prove for norm-2 but I haven't managed to do it for norm-infinity.
Thanks for your help.