  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.
