How to prove the existence of a fixed point of this mapping?

Question

Let $X=\{x_i : i\in I\}\subseteq\mathbf{R}^n$, where $I=\{1,\ldots,m\}$. Then for some initialization $\mu^{(t)}$, and $\pi^{(t)}=\{x\in X : \|x-\mu^{(t)}\|\leq r\}$, $r>0$, we want to prove that a mapping $\mu^{(t+1)}=M(\mu^{(t)})$ defined as $$ \mu^{(t+1)}=\textrm{argmin}_{\mu}\sum_{x\in \pi^{(t)}}\|x-\mu\| $$has a fixed point, i.e. there exist $\mu^*$ such that $\mu^*=M(\mu^*)$. We observe Euclidean norm.