Metrics and Quasimetrics Induced by Point Pair Function

Abstract

We study the point pair function in subdomains G of R-n. We prove that, for every domain G subset of R-n, this function is a quasi-metric with the constant less than or equal to root 5/2. Moreover, we show that it is a metric in the domain G = R-n\{0} with n >= 1. We also consider generalized versions of the point pair function, depending on a parameter alpha > 0, and show that in some domains these generalizations are metrics if and only if alpha <= 12.