Geometric properties of the Cassinian metric

Abstract

In this paper we prove a sharp distortion property of the Cassinian metric under Möbius transformations of a punctured ball onto another punctured ball. The paper also deals with discussion on local convexity properties of the Cassinian metric balls in some specific subdomains of ${\mathbb{R}}^n$. Inclusion properties of the Cassinian metric balls with other hyperbolic-type metric balls are also investigated. In particular, several conjectures are also stated in response to sharpness of the inclusion relations.

16 pages, 2 figures

References in corpus (5)

• Lipschitz conditions, triangular ratio metric, and quasiconformal maps
• Local Convexity Properties of Quasihyperbolic Balls in Punctured Space
• Geometry of the Cassinian metric and its inner metric
• Close-to-convexity of quasihyperbolic and $j$-metric balls
• Local Convexity Properties of j-metric Balls