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Feb 18, 2016 at 16:38 comment added Sebastian Goette From the description you give, the Karcher mean is continuous, at least in a weak sense: if a sequence of sets of $k$ elements converges to another set of $k$ elements, and if there exists a sequence of Karcher means that converges to a limit point, then that limit point is also a Karcher mean of the limit set.
Feb 18, 2016 at 14:22 vote accept nino
Feb 18, 2016 at 14:10 history answered Peter Michor CC BY-SA 3.0