Vol. 13, No. 1, 2020

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One-point hyperbolic-type metrics

Marina Borovikova, Zair Ibragimov, Miguel Jimenez Bravo and Alexandro Luna

Vol. 13 (2020), No. 1, 117–136
Abstract

We study basic properties of one-parametric families of the j̃-metric, the scale-invariant Cassinian metric and the half-Apollonian metric on locally compact, noncomplete metric spaces. We first establish basic properties of these metrics on once-punctured general metric spaces and obtain sharp estimates between these metrics, and then we show that all these properties, except for δ-hyperbolicity, extend to the settings of locally compact noncomplete metric spaces. We also show that these metrics are δ-hyperbolic only if the underlying space is a once-punctured metric space.

Keywords
semimetric spaces, metric spaces, Ptolemaic spaces, $\delta$-hyperbolic spaces, half-Apollonian metric, $\tilde j$-metric, scale-invariant Cassinian metric
Mathematical Subject Classification 2010
Primary: 30F45
Secondary: 51F99, 30C99
Milestones
Received: 17 June 2019
Revised: 3 October 2019
Accepted: 4 November 2019
Published: 4 February 2020

Communicated by Kenneth S. Berenhaut
Authors
Marina Borovikova
Department of Mathematics
California State University
Fullerton, CA
United States
Zair Ibragimov
Department of Mathematics
California State University
Fullerton, CA
United States
Miguel Jimenez Bravo
Department of Mathematics
California State University
Fullerton, CA
United States
Alexandro Luna
Department of Mathematics
California State University
Fullerton, CA
United States