In this article the authors study the Cassinian metric \(c_D\) which is defined in an arbitrary domain \(D\subsetneq \mathbb{R}^n\) by \[ c_D(x,y)=\sup_{p\in \partial D}\frac{|x-y|}{| x-p| | p-y|}. \] The distortion property of this metric under Möbius transformations of the unit ball has been studied already by Z. Ibragimov et al. [Bull. Malays. Math. Sci. Soc. (2) 40, No. 1, 361–372 (2017; Zbl 1366.30007)]. Thus in this paper one of the purposes is to prove the sharp distortion property of the Cassinian metric under Möbius maps from a punctured ball onto another punctured ball. The main theorem says that when \(a\in \mathbb{B}^n\) and \(f: \mathbb{B}^n\setminus \{0\} \to \mathbb{B}^n\setminus \{a\}\) is a Möbius map with \(f(0)=a,\) then for \(x,y\in \mathbb{B}^n\setminus\{0\}\) it holds that \[ \frac{1-|a|}{1+|a|}c_{_{\mathbb{B}^n{}^{\setminus\{0\}}}}(x,y)\leq c_{_{\mathbb{B}^n{}^{\setminus\{a\}}}}(f(x),f(y))\leq \frac{1+|a|}{1-|a|}c_{_{\mathbb{B}^n{}^{\setminus\{0\}}}}(x,y), \] and the equalities in both sides can be attained.
The second purpose of the paper is to focus on a general questions suggested by M. Vuorinen [in: Proceedings of the international workshop on quasiconformal mappings and their applications. New Delhi: Narosa Publishing House. 291–325 (2007; Zbl 1166.30013)] about the convexity of balls of small radii in metric spaces. Here are discussed the starlikeness and the convexity properties of the Cassinian metric balls.
The third purpose is to study inclusion properties of the Cassinian metric balls in proper subdomains \(D\) of \(\mathbb{R}^n\) with other related metric balls. To be more specific, for given \(x\in D \subsetneq \mathbb{R}^n\) and \(t>0\) the authors find optimal radii \(r,R>0\) depending only on \(x\) and \(t\) such that \[ B_d(x,r)\subset B_c(x,t)\subset B_d(x,R), \] where \(d\) is the other related metric defined on \(D,\) such as the Euclidean metric, the distance ratio metric, the hyperbolic or the quasihyperbolic metric. In particular, several conjectures are also stated in response to sharpness of the inclusion properties.