We settle an open problem of several years standing by showing that the
least-squares mean for positive definite matrices is monotone for the usual
(Loewner) order. Indeed we show this is a special case of its appropriate
generalization to partially ordered complete metric spaces of nonpositive
curvature. Our techniques extend to establish other basic properties of the
least squares mean such as continuity and joint concavity. Moreover, we
introduce a weighted least squares means and extend our results to this
setting.Comment: 21 page
