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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

IMPACT FACTOR 2018: 0.867

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SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

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Volume 27, Issue 5


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

Yongdo Lim / Miklós Pálfia
Published Online: 2013-11-05 | DOI: https://doi.org/10.1515/forum-2013-0088


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 t0+. 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.

Keywords: Hadamard space; barycenter; contractive mean; geometric power means

MSC: 53C23; 60B05; 53C20

About the article

Received: 2013-05-29

Revised: 2013-09-17

Published Online: 2013-11-05

Published in Print: 2015-09-01

Funding Source: National Research Foundation of Korea (NRF)

Award identifier / Grant number: 2012-005191

Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 2609–2635, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2013-0088.

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