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

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