Is there a name for subsets $X$ of the domain of a function $f$, especially $f \in SO(n,\Bbb R)$, such that $f(X)$ strictly contains $X$?

Question

I am asking for a friend who is interested in finding subsets $X$ of $\Bbb R^n$ such that for some rotation $r$ one has $ X \subsetneq r(X)$. Perhaps such things have been looked at and have a name? Any references would be appreciated. Of course, such subsets and rotations exist. But what are they called?