Forgive me if this question sounds silly.
Let $v$ be a $m \times 1$ vector and $P$ be a $m \times m$ permutation matrix. Can there be a transformation $T$ such that $\min_\limits{P\neq I}\|Tv-TPv\|$ exists?
Is this only a computational problem or are there analytic ideas about this problem, like invariant sets, etc.? Thanks!