Inequalities and bilipschitz conditions for triangular ratio metric

TL;DR

This paper compares the geometries induced by various metrics, including the triangular ratio metric, on domains in Euclidean space, and explores Lipschitz maps between these metric spaces.

Contribution

It provides inequalities and bilipschitz conditions relating the triangular ratio metric to other metrics, advancing understanding of their geometric relationships.

Findings

Derived inequalities between metrics

Established bilipschitz conditions for maps

Applied results to Lipschitz map analysis

Abstract

Let $G \subsetneq \mathbb{R}^n$ be a domain and let $d_1$ and $d_2$ be two metrics on $G$. We compare the geometries defined by the two metrics to each other for several pairs of metrics. The metrics we study include the distance ratio metric, the triangular ratio metric and the visual angle metric. Finally we apply our results to study Lipschitz maps with respect to these metrics.