Inequalities and bi-Lipschitz conditions for the triangular ratio metric

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.