Point pair function

We study intrinsic metrics in subdomains of the real $n$-dimensional Euclidean space $\mathbb{R}^n$, i.e. metrics which measure distances in the way that takes into account not only how close the points are to each other but also how the points are located with respect to the boundary of the domain (see, e.g. [1]). These metrics are often used to estimate the hyperbolic metric and, while they share some but not all of its properties, intrinsic metrics are much simpler than the hyperbolic metric and therefore more applicable.
Let $G$ be a proper subdomain of $\mathbb{R}^n$. Denote by $|x-z|$ the Euclidean distance in $\mathbb{R}^n$ and by $d_G(x)$ the distance from a point $x\in G$ to the boundary $\partial G$, i.e. $d_G(x):=\inf\{|x-z|\,:\,z\in\partial G\}$. We investigate the point pair function $p_G:G\times G\to[0,1)$ defined as
$$\tag{1} p_G(x,y)=\frac{|x-y|}{\sqrt{|x-y|^2+4d_G(x)d_G(y)}}\,,\quad x,y\in G. $$

We prove that for all domains $G\subsetneq\mathbb{R}^n$, the point pair function is a quasi-metric, i.e. it satisfies an analog of the triangle inequality with a multiplicative constant less than or equal to $\sqrt{5}/2$. Moreover, for $G=\mathbb{R}^n\backslash\{0\}$, $n\geq1$, this function defines a metric.
We also investigate what happens when the constant $4$ in (1) is replaced by another constant $\alpha>0$ to define a generalized version $p^\alpha_G$ of the point pair function $p_G$. In particular, we prove that, for $\alpha\in (0,12]$, this function $p^\alpha_G$ is a metric if $G$ is the positive real axis $\mathbb{R}^+$, the punctured space $\mathbb{R}^n\backslash\{0\}$ with $n\geq2$, or the upper half-space $\mathbb{H}^n$ with $n\geq2$. Furthermore, we also show that the function $p^\alpha_G$ is not a metric for any values of $\alpha>0$ in the unit ball $\mathbb{B}^n$.
This is a joint work with D. Dautova, O. Rainio, and M. Vuorinen [2].
References
[1] P. Hariri, R. Klén and M. Vuorinen, Conformally Invariant Metrics and Quasiconformal Mappings. Springer, 2020.
[2] D. Dautova, S. Nasyrov, O. Rainio and M. Vuorinen, "Metrics and quasimetrics induced by point pair function". Bulletin of the Brazilian Mathematical Society, New Series 2022.