Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ {$x\in X| d(x,C)<\epsilon$} and there are only finitely many $g\in G$ such that $g(U) \cap U \not= \emptyset$ ?

Cheers Helge