Geometry of the Cassinian metric and its inner metric

TL;DR

This paper explores the geometric properties of the Cassinian metric and its inner metric in Euclidean space, establishing inequalities and distortion properties under M"obius transformations.

Contribution

It provides new inequalities relating the Cassinian metric to other metrics and characterizes its distortion under M"obius transformations.

Findings

Derived inequalities between Cassinian and related metrics.

Established a sharp distortion property under M"obius transformations.

Analyzed the geometry of the Cassinian metric in specific subdomains.

Abstract

The Cassinian metric and its inner metric have been studied for subdomains of the $n$-dimensional Euclidean space $\mathbb{R}^n$ ($n\ge 2$) by the first named author. In this paper we obtain various inequalities between the Cassinian metric and other related metrics in some specific subdomains of $\mathbb{R}^n$. Also, a sharp distortion property of the Cassinian metric under M\"obius transformations of the unit ball is obtained.