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arXiv · 2202.08518

Metrics and quasimetrics induced by point pair function

Dina Dautova·Semen Nasyrov·Oona Rainio·Matti Vuorinen
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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 $\alpha\leq12$.

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Dina Dautova, Semen Nasyrov, Oona Rainio, Matti Vuorinen. 2022-02-17. Metrics and quasimetrics induced by point pair function. https://doi.org/10.1007/s00574-022-00309-5

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