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Geometric properties of the Cassinian metric

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 proper subdomains of Rn . 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.