Let $f : \mathbb{R}^n \to \mathbb{R}^{n}$ be an isometry. Is it true that $f(U) \subseteq U$ implies that $U \subseteq f(U)$ for an arbitrary subset $U$ of $\mathbb{R}^n$?
It is intuitively clear to me that $U$ and $f(U)$ must have the same "shape", so my guess is that if $f(U)$ is a subset of $U$ then the converse should also be true. I know it's a very simple statement but I haven't been able to prove it (so either my intuition is incorrect or I am missing something important). Any help would be really appreciated!