Intrinsic metrics defined with arithmetic and logarithmic mean values

Abstract

We introduce several new functions that measure the distance between two points $x$ and $y$ in a domain $G⊊{\mathbb{R}}^n$ 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.

15 pages