Generalizations of four hyperbolic-type metrics and Gromov hyperbolicity

TL;DR

This paper generalizes four hyperbolic-type metrics in metric spaces, proves their Gromov hyperbolicity, and shows the identity map is quasiconformal, extending known results and improving constants.

Contribution

It introduces new generalizations of hyperbolic metrics replacing boundary distance with Lipschitz functions, and establishes their Gromov hyperbolicity and quasiconformal properties.

Findings

All generalized metrics are Gromov hyperbolic spaces.

The identity map between original and generalized metrics is quasiconformal.

Improved Gromov constants for certain metrics.

Abstract

We study in the setting of a metric space $\left( X,d\right) $ some generalizations of four hyperbolic-type metrics defined on open sets $G$ with nonempty boundary in the $n-$dimensional Euclidean space, namely Gehring-Osgood metric, Dovgoshey- Hariri-Vuorinen metric, Nikolov-Andreev metric and Ibragimov metric. In the definitions of these generalizations, the boundary $\partial G$ of $G$ and the distance from a point $x$ of $G$ to $\partial G$ are replaced by a nonempty proper closed subset $M$ of $X$ and by a $1-$Lipschitz function positive on $X\setminus M$, respectively. For each generalization $\rho $ of the hyperbolic-type metrics mentioned above we prove that $\left( X\setminus M,\rho \right) $ is a Gromov hyperbolic space and that the identity map between $\left( X\setminus M,d\right) $ and $% \left( X\setminus M,\rho \right) $ is quasiconformal. For the Gehring-Osgood metric and the Nikolov-Andreev metric we improve the Gromov constants known from the literature. For Ibragimov metric the Gromov hyperbolicity is obtained even if we replace the distance from a point $x$ to $\partial G$ by any positive function on $X\setminus M$