Geometric properties of the Cassinian metric

Abstract

In this paper we prove a sharp distortion property of the Cassinian metric under M\"obius 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.Comment: 16 pages, 2 figure

Local convexity of metric balls

“…This paper may be considered to be a continuation of the earlier studies [HKLV,K1,K2,K3,KMS]. Our main results and their proofs suggest that similar results might be valid for other metrics as well and this offers ideas for further studies of the same topic, for instance for the Apollonian or the Seittenranta metrics [B2, S].…”

Mapping Properties of a Scale Invariant Cassinian Metric and a Gromov Hyperbolic Metric

“…Note that the quasihyperbolic and the distance ratio metrics do satisfy the bilipschitz property with bilipschitz constant 2 under Möbius maps (see [32, page 36], [8,Corollary 2.5] and [7,Proof of Theorem 4]). Similar properties have also been studied recently for the Cassinian metric [16] under Möbius maps of the unit ball and of a punctured ball onto another punctured ball (see [19,22]).…”