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 $\stackrel{̃}{j}$-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 $\delta$-hyperbolicity, extend to the settings of locally compact noncomplete metric spaces. We also show that these metrics are $\delta$-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