Inequalities and bilipschitz conditions for triangular ratio metric

Abstract

Let $G⊊{\mathbb{R}}^n$ 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.

21 pages, 3 figures

References in corpus (4)

• Dimensions, Whitney covers, and tubular neighborhoods
• Lipschitz conditions, triangular ratio metric, and quasiconformal maps
• Some remarks on the visual angle metric
• The visual angle metric and quasiregular maps

Cited by in corpus (14)

• Some remarks on the Cassinian metric
• Intrinsic quasi-metrics
• Barrlund's distance function and quasiconformal maps
• Triangular ratio metric under quasiconformal mappings in sector domains
• Triangular ratio metric in the unit disk
• Intrinsic metrics under conformal and quasiregular mappings
• Some remarks on the visual angle metric
• Metrics and quasimetrics induced by point pair function
• Introducing a new intrinsic metric
• Möbius metric in sector domains
• The Ptolemy-Alhazen problem and spherical mirror reflection
• The Ptolemy-Alhazen problem and quadric surface mirror reflection
• Intrinsic metrics in ring domains
• A Gromov Hyperbolic metric and Möbius transformations