First of all sorry for the clunchy title, I had no idea how to phrase it better.
Take $M$ a Riemann connected compact manifold with induced distance $d$, let $\phi:M\to M$ be an isometry. We define $p_0$ as the point minimizing $M\to\mathbb{R}:p\mapsto d(p,\phi(p))$
Is it true that there exists a minimizing geodesic that goes from $p_0$ to $\phi(p_0)$?
It's part of an exercice I'm trying to do, so even some suggestions from where I can work on will be very appreciated.
I have the feeling that, as we take $p_0$ in this way, it's possible to show that $\phi(p_0)$ is in the image of the exponential map of $p_0$, and therefore there is a minimmizing geodesic. But I don't know how to prove it (if it's true).