Necessary and sufficient condition for the existence of a Fr\'echet mean on the circle

Let $(\S^1,d_{\S^1})$ be the unit circle in $\R^2$ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure $\mu$ to admit a well defined Fr\'echet mean on $(\S^1,d_{\S^1})$. %This criterion allows to recover already known sufficient conditions of existence. We derive a new sufficient condition of existence $P(\alpha,\varphi)$ with no restriction on the support of the measure. Then, we study the convergence of the empirical Fr\'echet mean to the Fr\'echet mean and we give an algorithm to compute it.