Geometric properties of a new hyperbolic type metric
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915461956894720 |
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| author | Chen, Xinyu Zhang, Xiaohui |
| author_facet | Chen, Xinyu Zhang, Xiaohui |
| contents | A new distance function $\tilde{S}_{G,c}$ in metric space $(X,d)$ is introduced as \begin{align*} &\tilde{S}_{G,c}(x,y)=\log{\left(1+\frac{cd(x,y)}{\sqrt{1+d(x)}\sqrt{1+d(y)}}\right)} \end{align*} for $x$, $y\in X$ and $c$ is an arbitrary positive real number. We find that $\tilde{S}_{G,c}$ is a metric for $c\ge 2$. In general, the condition $c\geq2$ can not be improved. In this paper we investigate some geometric properties of the metric $\tilde{S}_{G,c}$ including the comparison inequalities between this metric and the triangular ratio metric and the inclusion relation between some metric balls. We show the quasiconformality of a bilipschitz mapping in metric $\tilde{S}_{G,c}$ and the distortion property of the metric $\tilde{S}_{\partial\mathbb{B}^n,c}$ under Möbius transformations of the unit ball. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_17956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric properties of a new hyperbolic type metric Chen, Xinyu Zhang, Xiaohui Metric Geometry 30F45, 51M05 A new distance function $\tilde{S}_{G,c}$ in metric space $(X,d)$ is introduced as \begin{align*} &\tilde{S}_{G,c}(x,y)=\log{\left(1+\frac{cd(x,y)}{\sqrt{1+d(x)}\sqrt{1+d(y)}}\right)} \end{align*} for $x$, $y\in X$ and $c$ is an arbitrary positive real number. We find that $\tilde{S}_{G,c}$ is a metric for $c\ge 2$. In general, the condition $c\geq2$ can not be improved. In this paper we investigate some geometric properties of the metric $\tilde{S}_{G,c}$ including the comparison inequalities between this metric and the triangular ratio metric and the inclusion relation between some metric balls. We show the quasiconformality of a bilipschitz mapping in metric $\tilde{S}_{G,c}$ and the distortion property of the metric $\tilde{S}_{\partial\mathbb{B}^n,c}$ under Möbius transformations of the unit ball. |
| title | Geometric properties of a new hyperbolic type metric |
| topic | Metric Geometry 30F45, 51M05 |
| url | https://arxiv.org/abs/2508.17956 |