Metrics and quasimetrics induced by point pair function

TL;DR

This paper investigates the properties of the point pair function in various domains, establishing conditions under which it acts as a quasi-metric or metric, and exploring its generalizations with different parameters.

Contribution

It proves that the point pair function is a quasi-metric with a specific constant in all proper subdomains of Euclidean space and characterizes when its generalizations are metrics.

Findings

The point pair function is a quasi-metric with constant ≤ √5/2 in all proper subdomains.

It is a true metric in the punctured Euclidean space for all dimensions.

Generalized point pair functions are metrics if and only if the parameter α ≤ 12.

Abstract

We study the point pair function in subdomains $G$ of $\mathbb{R}^n$. We prove that, for every domain $G\subsetneq\mathbb{R}^n$, 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 $n\geq1$. 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 $\alpha\leq12$.