  We introduce several new functions that measure the distance between two
points $x$ and $y$ in a domain $G\subsetneq\mathbb{R}^n$ by using the
arithmetic or the logarithmic mean of the Euclidean distances from the points
$x$ and $y$ to the boundary of $G$. We study in which domains these functions
are metrics and find sharp inequalities between them and the hyperbolic metric.
We also present one result about their distortion under quasiregular mappings.
