Geometric properties of a new hyperbolic type metric

Xinyu Chen; Xiaohui Zhang

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arXiv · 2508.17956

Geometric properties of a new hyperbolic type metric

Xinyu Chen·Xiaohui Zhang

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Abstract

We introduce a new distance function $\tilde{S}_{G,c}$ in a metric space $(X,d)$, defined by \[ \tilde{S}_{G,c}(x,y)=\log\left(1+\frac{c\,d(x,y)}{\sqrt{1+d(x,G)}\sqrt{1+d(y,G)}}\right) \] for $x, y \in X$, where $c$ is a positive real number and $d(x,G)$ denotes the distance from $x$ to the subset $G \subset X$. We establish that $\tilde{S}_{G,c}$ is a metric for $c \geq 2$. Moreover, we show that the condition $c \geq 2$ is sharp. This paper investigates geometric properties of the metric $\tilde{S}_{G,c}$, including comparison inequalities with the triangular ratio metric and inclusion relations among metric balls. We demonstrate the quasiconformality of bilipschitz mappings with respect to $\tilde{S}_{G,c}$ and study the distortion property of the metric $\tilde{S}_{\partial\mathbb{B}^{n},c}$ under Möbius transformations of the unit ball.

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BibTeX RIS

Xinyu Chen, Xiaohui Zhang. 2026-07-28. Geometric properties of a new hyperbolic type metric. https://arxiv.org/abs/2508.17956

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