Abstract: The Fréchet mean generalizes the concept of a mean to metric space settings. In this talk we consider equivariant estimation of Fréchet means with respect to the isometry group action on a Riemannian manifold. For some models, there exists an optimal equivariant estimator, which will necessarily perform as well or better than other common equivariant estimators, such as the maximum likelihood estimator or the sample Fréchet mean. We derive the general form of this minimum risk equivariant estimator in terms of Haar measures and in some cases provide explicit expressions for it. A result for finding the Fréchet mean for distributions with radially decreasing densities is presented and used to find expressions for the minimum risk equivariant estimator. Simulation results show that the minimum risk equivariant estimator performs favourably relative to alternative estimators.