P. Hariri et al. [Conformally invariant metrics and quasiconformal mappings. Cham: Springer (2020; Zbl 1450.30003)] have singled out properties of the hyperbolic metric (one of them being the property that closures of the balls never intersect the boundary of the domain on which they are defined) and have called generalizations of the hyperbolic metric that fulfills those properties hyperbolic type metrics. Any metric whose values are affected by the boundary of the domain is referred to as an intrinsic metric, regardless whether it has also the other properties of a hyperbolic type metric or not.
This paper introduces a new intrinsic metric, called the \(t\)-metric, proves sharp inequalities between it and the most common hyperbolic type metrics, for several domains \(G\subseteq {\mathbb R}^n\), studies its behaviour under a few examples of conformal and quasiconformal mappings, and looks, by omputational and analytical means, at the differences between the balls defined by the metrics considered.
The \(t\)-metric is defined on any non-empty, open, proper and connected subset \(G\) of a metric space \(X\). Given a metric \(\eta_G\) defined in the closure of \(G\) and denoting \(\eta_G(x) = \eta_G(x, \partial G) = \inf\{\eta_G(x, z)\, |\, z \in \partial G\}\) for all \(x\in G\), one lets \(t_G : G\times G\rightarrow [0, 1]\) be defined, for all \(x, y\in G\), by \[ t_G(x, y) =\frac{\eta_G(x, y)}{\eta_G(x, y) + \eta_G(x) + \eta_G(y)}. \] The focus is on \(t\)-metrics associated with \(G\subseteq {\mathbb R}^n\) with \(\eta_G\) being the Euclidean distance. The \(t\)-metric does not have the property about the closed balls not intersecting with the boundary, so it is not a hyperbolic type metric.