Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not unitarily invariant obviously). How can we choose $U$ in the unitary matrices to minimize $g$? Or what kind of results are there regarding the minimum value?

I'm particularly interested in $||.||_{\infty}$, the tail expectation, and maybe also the quantile function (i.e. $||x||$ is the $k$th largest element ... which will not be a norm.)

I'm mostly looking to "bind" this problem in the right way so that I can read the most appropriate material. I would imagine this kind of problem has been studied to death in various contexts. Or maybe I'm not seeing that the problem is in fact trivial or I've mis-specified it.