Necessary and sufficient condition for the existence of a Fréchet 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 also derive a new sufficient condition without restriction on the support of the measure. Then, we study the convergence of the empirical Fr\'echet mean to the Fr\'echet mean. An algorithm to compute the empirical Fr\'echet mean is also given.