Abstract
We study basic properties of one-parametric families of the -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 -hyperbolicity, extend to the settings of locally compact noncomplete metric spaces. We also show that these metrics are -hyperbolic only if the underlying space is a once-punctured metric space.
Citation
Marina Borovikova. Zair Ibragimov. Miguel Jimenez Bravo. Alexandro Luna. "One-point hyperbolic-type metrics." Involve 13 (1) 117 - 136, 2020. https://doi.org/10.2140/involve.2020.13.117
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