Averaging one-point hyperbolic-type metrics

TL;DR

This paper demonstrates that averaging certain one-point hyperbolic-type metrics preserves Gromov hyperbolicity, unlike taking the supremum, and shows the hyperbolicity constant remains unaffected by the number of metrics averaged.

Contribution

It introduces an averaging method for one-point hyperbolic-type metrics that maintains Gromov hyperbolicity, contrasting with the supremum approach.

Findings

Averaging preserves Gromov hyperbolicity of certain metrics.

The hyperbolicity constant is independent of the number of metrics averaged.

The average of hyperbolic metrics may not always be hyperbolic.

Abstract

It is known that the $\tilde j$-metric, half-apollonian metric and scale-invariant Cassinian metric are not Gromov hyperbolic. These metrics are defined as a supremum of one-point metrics (i.e., metrics constructed using one boundary point) and the supremum is taken over all boundary points. The aim of this paper is to show that taking the average instead of the supremum yields a metric that preserves the Gromov hyperbolicity. Moreover, we show that the Gromov hyperbolicity constant of the resulting metric does not depend on the number of metrics used in taking the average. We also provide an example to show that the average of Gromov hyperbolic metrics is not, in general, Gromov hyperbolic.