Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define
$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)), $$
that is, $r(p)$ is the minimum distance to $p$ of a point in its orbit. As the sphere is a homogeneous manifold, I expect $r$ to be a constant function, although I couldn't prove it. Any thoughts on it?