Metrization of the Gromov–Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces

Khezeli, Ali

doi:10.1016/j.spa.2019.11.001

Cited by 6 publications

(30 citation statements)

References 17 publications

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“…Such limits are random continuum metric spaces and can be defined by weak convergence w.r.t. the Gromov-Hausdorff-Prokhorov metric [37]. It is shown in the preprint [6] that if the unimodular discrete space admits a scaling limit, then the ordinary Hausdorff dimension of the limit is an upper bound for the unimodular Hausdorff dimension.…”

Section: Max-flow Min-cut Theorem For Unimodular One-ended Treesmentioning

confidence: 99%

Unimodular Hausdorff and Minkowski dimensions

Baccelli

Haji-Mirsadeghi

Khezeli

2021

Electron. J. Probab.

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This work introduces two new notions of dimension, namely the unimodular Minkowski and Hausdorff dimensions, which are inspired from the classical analogous notions. These dimensions are defined for unimodular discrete spaces, introduced in this work, which provide a common generalization to stationary point processes under their Palm version and unimodular random rooted graphs. The use of unimodularity in the definitions of dimension is novel. Also, a toolbox of results is presented for the analysis of these dimensions. In particular, analogues of Billingsley's lemma and Frostman's lemma are presented. These last lemmas are instrumental in deriving upper bounds on dimensions, whereas lower bounds are obtained from specific coverings. The notions of unimodular Hausdorff size, which is a discrete analogue of the Hausdorff measure, and unimodular dimension function are also introduced. This toolbox allows one to connect the unimodular dimensions to other notions such as volume growth rate, discrete dimension and scaling limits. It is also used to analyze the dimensions of a set of examples pertaining to point processes, branching processes, random graphs, random walks, and self-similar discrete random spaces. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.

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Section: Max-flow Min-cut Theorem For Unimodular One-ended Treesmentioning

confidence: 99%

Unimodular Hausdorff and Minkowski dimensions

Baccelli

Haji-Mirsadeghi

Khezeli

2021

Electron. J. Probab.

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“…By a method similar to the one used in Abraham, Delmas and Hoscheit [1] (or Khezeli [18]), one can extend the Gromov-Hausdorff-Prokhorov distance to pointed forward Θ-metric-measure spaces. In particular, the corresponding topology is equivalent to the measured forward Θ-Gromov-Hausdorff topology.…”

Section: Example 32 (A Flaw Gromov-hausdorff Distance)mentioning

confidence: 99%

On the geometry of irreversible metric-measure spaces: Convergence, Stability and Analytic aspects

Kristály

Zhao

2021

Preprint

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The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable -reversibility depending -Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.

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“…As mentioned in the introduction, the Gromov-Hausdorff (-Prokhorov) metric is also defined for boundedly-compact pointed (measured) metric spaces as well (see [21] and also [1] and [5]). This metric generates the Gromov-Hausdorff (-Prokhorov) topology on N * (resp.…”

Section: The Gromov-hausdorff (-Prokhorov) Metricmentioning

confidence: 99%

“…The notion of Gromov-Hausdorff-Prokhorov convergence is defined similarly on the set M * of boundedly-compact pointed measured metric spaces [25] (also called measured Gromov-Hausdorff convergence). It is known that these topologies are metrizable and N * and M * become complete and separable metric spaces (this was shown for length spaces and discrete metric spaces in [1] and [5] respectively and the general case is done in [21]). This enables one to study random (measured) non-compact metric spaces.…”

Section: Introductionmentioning

confidence: 99%

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A Unified Framework for Generalizing the Gromov-Hausdorff Metric

Khezeli¹

2018

Preprint

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In this paper, a general approach is presented for generalizing the Gromov-Hausdorff metric to consider metric spaces equipped with some additional structure. A special case is the Gromov-Hausdorff-Prokhorov metric which considers measured metric spaces. This abstract framework also unifies several existing generalizations which consider metric spaces equipped with a measure, a point, a closed subset, a curve or a tuple of such structures. It can also be useful for studying new examples of additional structures. The framework is provided both for compact metric spaces and for boundedly-compact pointed metric spaces. In addition, completeness and separability of the metric is proved under some conditions. This enables one to study random metric spaces equipped with additional structures, which is the main motivation of this work.

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Metrization of the Gromov–Hausdorff (-Prokhorov) topology for boundedly-compact metric spaces

Cited by 6 publications

References 17 publications

Unimodular Hausdorff and Minkowski dimensions

Unimodular Hausdorff and Minkowski dimensions

On the geometry of irreversible metric-measure spaces: Convergence, Stability and Analytic aspects

A Unified Framework for Generalizing the Gromov-Hausdorff Metric

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