Lipschitz conditions, triangular ratio metric, and quasiconformal maps

Abstract

The triangular ratio metric is studied in subdomains of the complex plane and Euclidean $n$-space. Various inequalities are proven for it. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii.

30 pages, four figures

References in corpus (2)

• Geometry of Metrics
• Weak metrics on Euclidean domains

Cited by in corpus (19)

• Inequalities and bilipschitz conditions for triangular ratio metric
• Comparison theorems for hyperbolic type metrics
• Geometry of the Cassinian metric and its inner metric
• Some remarks on the Cassinian metric
• Triangular ratio metric under quasiconformal mappings in sector domains
• Intrinsic quasi-metrics
• Barrlund's distance function and quasiconformal maps
• Triangular ratio metric in the unit disk
• Intrinsic metrics under conformal and quasiregular mappings
• Some remarks on the visual angle metric
• Geometric properties of the Cassinian metric
• Metrics and quasimetrics induced by point pair function
• Introducing a new intrinsic metric
• Möbius metric in sector domains
• Hilbert metric in the unit ball
• Hyperbolic type distances in starlike domains
• A Gromov Hyperbolic metric and Möbius transformations
• Intrinsic metrics in ring domains
• The Ptolemy-Alhazen problem and quadric surface mirror reflection