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Averaging one-point hyperbolic-type metrics


Authors: Asuman Güven Aksoy, Zair Ibragimov and Wesley Whiting
Journal: Proc. Amer. Math. Soc. 146 (2018), 5205-5218
MSC (2010): Primary 30F45; Secondary 51F99, 30C99
DOI: https://doi.org/10.1090/proc/14173
Published electronically: September 4, 2018
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Abstract: It is known that the $ \tilde \jmath $-metric, the half-Apollonian metric, and the scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point), and the supremum is taken over all boundary points. The aim of this paper is to show that taking the average instead of the supremum yields a metric that is Gromov hyperbolic. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of boundary points used in taking the average. We also provide an example to show that the average of Gromov hyperbolic metrics is not, in general, Gromov hyperbolic.


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Additional Information

Asuman Güven Aksoy
Affiliation: Department of Mathematics, Claremont McKenna College, Claremont, California 91711
Email: aaksoy@cmc.edu

Zair Ibragimov
Affiliation: Department of Mathematics, California State University at Fullerton, Fullerton, California 92831
Email: zibragimov@fullerton.edu

Wesley Whiting
Affiliation: Department of Mathematics, California State University at Fullerton, Fullerton, California 92831
Email: weswhiting@gmail.com

DOI: https://doi.org/10.1090/proc/14173
Keywords: Averages, Cassinian metrics, one-point metrics, hyperbolic-type metrics, Gromov hyperbolicity
Received by editor(s): September 12, 2017
Received by editor(s) in revised form: February 5, 2018, and March 22, 2018
Published electronically: September 4, 2018
Communicated by: Jeremy T. Tyson
Article copyright: © Copyright 2018 American Mathematical Society

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