On the Gromov-Hausdorff quasi-metric distance
Keywords:
Hausdorff distance, Gromov-Hausdorff distance, Isbell-convex quasi-metric spaceAbstract
Dedicated to Professor Rebecca Walo Omana on the occasion of her promotion to emeritus professor
We introduce the concept of the Gromov-Hausdorff quasi-metric distance between two quasi-metric spaces. We then use this concept to study the stability estimates of two Isbell-hulls quasi-metric spaces. Moreover, we obtain the asymmetric version of the following well-known result: the Gromov-Hausdorff distance of two hyperconvex metric spaces generated by certain subsets is less than or equal to the Gromov-Hausdorff distance of these sets.
References
D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, AMS, 2001.
S. Cobzas, Functional analysis in Asymmetric spaces, Frontiers in Mathematics, Springers, Basel. (2013).
J. Conradie, H.-P. A. Künzi and O. Olela Otafudu, The vector lattice structure on the Isbell-convex hull of an asymmetrically normed real vector space, Topology Appl. 231 (2017), 92-112.
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci. Publ. Math. 53 (1981), 53-73.
E. Kemajou, H.-P. Kunzi, and O. Olela Otafudu, The Isbell-Hull of a di-space, Topology Appl. 159 (2012), 2463-2475.
H.-P.A. Künzi, An introduction to quasi-uniform spaces, Contemp. Math. 486 (2009), 239-304.
H.-P. Künzi, M. Sanchis, The Katětov construction modified for a $T_0$-quasimetric space, Topology Appl. 159 (2012), 711-720.
H.-P. Künzi and O.Olela Otafudu, The ultra-quasi-metrically injective hull of a $T_0$-ultra-quasi-metric space, Appl. Categ. Structures, 21 (2013), 651-670.
U. Lang, M. Pavón, R. Züst, Metric stability of trees and tight spans, Arch. Math. 101 (2013), 91-100.
S. Lim, F. Mémoli and Z. Smith, The Gromov-Hausdorff distance between spheres, Geom. Topol. 27 (2023), 3733-3800.
F. Mémoli, S. Facundo and W. Zane, Zhengchao, The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation, J. Comput. Geom. 14 (2023), 78-143.
F. Mémoli, Gromov-Wasserstein distances and the metric approach to object matching, Found. Comput. Math. 11 (2011), 417-487.
A. Moezzi, The Injective Hull of Hyperbolic Groups, Dissertation ETH Zurich, No. 18860, 2010.
O. Olela-Otafudu, Gated sets in $T_0$-quasi-metric spaces, Topology Appl. 263 (2019), 159-171.
O. Olela-Otafudu, The injective hull of ultra-quasi-metric versus $q$-hyperconvex hull of quasi-metric space, Topology Appl. 203 (2016), 170-176.
O. Olela-Otafudu, Extremal function pairs in asymmetrically normed linear spaces, Topology Appl. 166 (2014), 98-107.
D. Qiu, Geometry of non-Archimedean Gromov-Hausdorff distance, p-Adic Numbers, Ultrametric Anal., Appl. 1 (2009), 317-337.
Y . Shi and X . Wei, On stability of ultrametrically injective hulls, Topology Appl. 370 (2025), 109437.
