Metrics and quasimetrics induced by point pair function

Abstract

We study the point pair function in subdomains $G$ of ${\mathbb{R}}^n$. We prove that, for every domain $G⊊{\mathbb{R}}^n$, the this function is a quasi-metric with the constant less than or equal to $\sqrt{5}\slash2$. Moreover, we show that it is a metric in the domain $G={\mathbb{R}}^n\setminus \{0\}$ with $n\ge 1$. We also consider generalized versions of the point pair function, depending on an arbitrary constant $\alpha >0$, and show that in some domains these generalizations are metrics if and only if $\alpha \le 12$.

21 pages, 3 figures

References in corpus (3)

• Triangular ratio metric under quasiconformal mappings in sector domains
• Triangular ratio metric in the unit disk
• Intrinsic quasi-metrics