Gromov hyperbolic John is quasihyperbolic John II

TL;DR

This paper introduces quasihyperbolic John spaces with centers, provides criteria for identifying such spaces, and proves that Gromov hyperbolic quasihyperbolic John spaces are indeed quasihyperbolic John, answering an open question.

Contribution

It defines quasihyperbolic John spaces with centers and proves a key property linking Gromov hyperbolic quasihyperbolic John spaces to quasihyperbolic John spaces.

Findings

Established criteria for quasihyperbolic John spaces.

Proved that Gromov hyperbolic quasihyperbolic John spaces are quasihyperbolic John.

Answered an open question by Heinonen (1989).

Abstract

In this paper, we introduce a concept of quasihyperbolic John spaces (with center) and provide a criteria to determine spaces to be quasihyperbolic John. As an application, we provide a simple proof to show that a John space with a Gromov hyperbolic quasihyperbolization is quasihyperbolic John, quantitatively. This gives an affirmative answer to an open question posed by Heinonen (Rev.~Math.~Iber, 1989), which has been studied by Gehring et al. (Math.~Scand, 1989).