Approximations to the Karcher mean on Hadamard spaces via geometric power means

In this paper we establish a limit theorem for contractive means on a Hadamard space. For a contractive mean satisfying an extended metric inequality, the corresponding geometric power means varying over t∈(0,1] converge to the Karcher mean (center of gravity or Cartan mean) as t→0+. This provides not only a deterministic approach to the Karcher mean, but also a simple proof of some important properties of the Karcher mean, which have been established by Sturm using probabilistic methods on the metric structure of Hadamard spaces. We obtain further new results for the Karcher mean via this deterministic method, especially we derive a characteristic property of the Karcher mean and effective and explicit upper bounds for distances between Karcher means.