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We formulate a variational problem on a Riemannian manifold M whose solutions are piecewise smooth geodesics that best fit a given data set of time labelled points in M . By a limiting process, these solutions converge to a single point in M , that we prove to be the Riemannian mean of the given points for some particular Riemannian manifolds such as Euclidean spaces, connected and compact Lie groups and spheres.

Riemannian Means as Solutions of Variational Problems