Geometric properties of a new hyperbolic type metric

TL;DR

This paper introduces a new hyperbolic-type metric in metric spaces, explores its geometric properties, and examines its behavior under transformations, providing insights into its structure and applications.

Contribution

The paper defines a new metric $ ilde{S}_{G,c}$, proves it is valid for $c extgreater 2$, and investigates its geometric properties and transformation behaviors.

Findings

$ ilde{S}_{G,c}$ is a metric for $c extgreater 2$

Comparison inequalities with the triangular ratio metric

Inclusion relations between metric balls and quasiconformal mappings

Abstract

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 M\"obius transformations of the unit ball.