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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 30, Issue 3


The quasi-arithmetic means and Cartan barycenters of compactly supported measures

Sejong Kim
Published Online: 2017-10-06 | DOI: https://doi.org/10.1515/forum-2017-0136


Since positive definite Hermitian matrices have become fundamental objects in many areas, a variety of theoretical and computational research topics have been arisen. Especially, the average of positive definite matrices is a very important notion to see the central tendency of objects. There are many different kinds of averages for a finite number of positive definite matrices such as quasi-arithmetic means, power means and Cartan barycenters. We generalize these averages to the setting of positive definite matrices equipped with probability measures of compact support, and show the monotonicity of quasi-arithmetic means for parameters 1, and connections with inequalities between quasi-arithmetic means and power means, and between quasi-arithmetic means and Cartan barycenters.

Keywords: Compactly supported measure; contraction property; quasi-arithmetic mean; power mean,Cartan barycenter

MSC 2010: 28A25; 15B48


  • [1]

    E. Ahn, S. Kim and Y. Lim, An extended Lie–Trotter formula and its applications, Linear Algebra Appl. 427 (2007), no. 2–3, 190–196. Web of ScienceCrossrefGoogle Scholar

  • [2]

    V. Arsigny, P. Fillard, X. Pennec and N. Ayache, Geometric means in a novel vector space structure on symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl. 29 (2006/2007), no. 1, 328–347. Web of ScienceGoogle Scholar

  • [3]

    M. Berger, A Panoramic View of Riemannian Geometry, Springer, Berlin, 2003. Google Scholar

  • [4]

    R. Bhatia and J. Holbrook, Riemannian geometry and matrix geometric means, Linear Algebra Appl. 413 (2006), no. 2–3, 594–618. CrossrefGoogle Scholar

  • [5]

    J. Bochi and A. Navas, A geometric path from zero Lyapunov exponents to rotation cocycles, Ergodic Theory Dynam. Systems 35 (2015), no. 2, 374–402. CrossrefGoogle Scholar

  • [6]

    V. I. Bogachev, Measure Theory. Vol. I, II, Springer, Berlin, 2007. Google Scholar

  • [7]

    F. Bolley, Separability and completeness for the Wasserstein distance, Séminaire de probabilités XLI, Lecture Notes in Math. 1934, Springer, Berlin (2008), 371–377. Google Scholar

  • [8]

    P. S. Bullen, Handbook of Means and Their Inequalities, Math. Appl. 560, Kluwer Academic Publishers, Dordrecht, 2003. Google Scholar

  • [9]

    T. Champion, L. De Pascale and P. Juutinen, The -Wasserstein distance: Local solutions and existence of optimal transport maps, SIAM J. Math. Anal. 40 (2008), no. 1, 1–20. Google Scholar

  • [10]

    F. Hansen, J. Pečarić and I. Perić, Jensen’s operator inequality and its converses, Math. Scand. 100 (2007), no. 1, 61–73. CrossrefGoogle Scholar

  • [11]

    F. Hiai and Y. Lim, Log-majorization and Lie–Trotter formula for the Cartan barycenter on probability measure spaces, J. Math. Anal. Appl. 453 (2017), no. 1, 195–211. Web of ScienceCrossrefGoogle Scholar

  • [12]

    R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. Google Scholar

  • [13]

    H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), no. 5, 509–541. CrossrefGoogle Scholar

  • [14]

    S. Kim and H. Lee, The power mean and the least squares mean of probability measures on the space of positive definite matrices, Linear Algebra Appl. 465 (2015), 325–346. CrossrefWeb of ScienceGoogle Scholar

  • [15]

    F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1979/80), no. 3, 205–224. Google Scholar

  • [16]

    J. D. Lawson and Y. Lim, Contractive barycentric maps, J. Operator Theory 77 (2017), no. 1, 87–107. Web of ScienceCrossrefGoogle Scholar

  • [17]

    Y. Lim and M. Pálfia, Matrix power means and the Karcher mean, J. Funct. Anal. 262 (2012), no. 4, 1498–1514. Web of ScienceCrossrefGoogle Scholar

  • [18]

    Y. Lim and T. Yamazaki, On some inequalities for the matrix power and Karcher means, Linear Algebra Appl. 438 (2013), no. 3, 1293–1304. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris 2002), Contemp. Math. 338, American Mathematical Society, Providence (2003), 357–390. Google Scholar

  • [20]

    A. C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc. 14 (1963), 438–443. Google Scholar

  • [21]

    C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, American Mathematical Society, Providence, 2003. Google Scholar

About the article

Received: 2017-06-27

Revised: 2017-09-12

Published Online: 2017-10-06

Published in Print: 2018-05-01

Funding Source: National Research Foundation of Korea

Award identifier / Grant number: 2015R1C1A1A02036407

This work was supported by a Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT and Future Planning (2015R1C1A1A02036407).

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 753–765, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0136.

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