We introduce several new functions that measure the distance between two
points x and y in a domain GβRn 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.Comment: 15 page