Suppose a discrete group $G$ acts properly and isometrically on a Riemannian manifold $M$. Denote the Riemannian distance function by $d$.
Pick a point $x\in M$. For each $x'\in M$, consider the subset $N_{x'}$ of points in the orbit $G\cdot x'$ that are closest to $x$:
$$N_{x'}:=\{y\in G\cdot x':d(x,y)=d(x,N_{x'})\}.$$
Let $N:=\bigcup_{x'\in X}N_{x'}.$
Question: Does there exist a constant $C$ such that for any $z,z'\in N$, we have $$d(z,z')\leq d(G\cdot z,G\cdot z')+C?$$
Comment: It is clear that $C$ cannot always be $0$, since $z$ and $z'$ may be two points in the same orbit of equal and minimal distance to $x$. But it's not clear to me whether such a $C$ always exists.