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Monotonic properties of the least squares mean

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Abstract

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 mean and derive our results in this more general setting.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Jimmie Lawson

  2. Department of Mathematics, Kyungpook National University, Taegu, 702-701, Korea

    Yongdo Lim

Authors
  1. Jimmie Lawson
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  2. Yongdo Lim
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Correspondence to Jimmie Lawson.

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Lawson, J., Lim, Y. Monotonic properties of the least squares mean. Math. Ann. 351, 267–279 (2011). https://doi.org/10.1007/s00208-010-0603-6

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  • DOI: https://doi.org/10.1007/s00208-010-0603-6

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