Geometry of the Cassinian metric and its inner metric

The Cassinian metric and its inner metric have been studied for subdomains of the $n$-dimensional Euclidean space ${\mathbb{R}}^n$ ($n\ge 2$) by the first named author. In this paper we obtain various inequalities between the Cassinian metric and other related metrics in some specific subdomains of ${\mathbb{R}}^n$. Also, a sharp distortion property of the Cassinian metric under M"obius transformations of the unit ball is obtained.