Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that $g(x_0)\neq x_0$ when $g\neq\mathrm{id}$ and a set $$D=\{x\in X\mid \mathrm d(x,x_0)\leq\mathrm d(x,g(x_0))\quad\forall g\in G\}.$$
Is it true that $\mathrm{int}(D)$ contains at most one point out of each orbit? A possibility would be $$\mathrm{int}(D)\overset?=\{x\in X\mid \mathrm d(x,x_0)<\mathrm d(x,g(x_0))\quad\forall g\in G\setminus\{\mathrm{id}\}\}.$$ And is $\mathrm{int}(D)\neq\varnothing$?