Thanks. Based on your argument, I will use a slightly tighter bound $1+2a_{\max}\cos\frac{\pi}{n+1}$ where $a_{\max} = \max_{i=1\ldots n-1}a_{i}$.

I have heard that this problem is a popular optimization interview question.

Thank you for taking your time and writing a full solution with clear instructions.

Are you sure about the first constraint? It just gives you the solution for x as $x=\Phi v$. Also, can you specify the dimensions of $A$ and $\Phi$?

Well, I am pretty sure someone should have derived the concentration inequalities for heavy-tail distributions. I tried to look at this: Robust Estimation of High-Dimensional Mean Regression, but their lemmas require me to read the entire paper to understand. I'll look into the robust statistic literature to see what I can find.