Stability of the q-hyperconvex hull of a quasi-metric space

TL;DR

This paper investigates the stability of the q-hyperconvex hull in quasi-metric spaces by extending metric concepts like Gromov-Hausdorff distance, showing that similar spaces have close hyperconvex hulls.

Contribution

It extends stability results and metric notions from metric to quasi-metric spaces, providing new insights into their hyperconvex hulls.

Findings

q-hyperconvex hulls are close if the original spaces are close

Extended Gromov-Hausdorff distance to quasi-metric spaces

Characterized spaces that are Sym-large in their hyperconvex hulls

Abstract

In this paper, we study the stability of the q-hyperconvex hull of a quasi-metric space, adapting known results for the hyperconvex hull of a metric space. To pursue this goal, we extend well-known metric notions, such as Gromov-Hausdorff distance and rough isometries, to the realm of quasi-metric spaces. In particular, we prove that two q-hyperconvex hulls are close with respect to the Gromov-Hausdorff distance if so are the original spaces. Moreover, we provide an intrinsic characterisation of those spaces that are Sym-large in their q-hyperconvex hulls.