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The Gromov-Hausdorff distance between $l^p$-products of metric spaces

Emin Abdullaev

Abstract

This paper studies $l^p$-products of metric spaces and provides estimates for the Gromov-Hausdorff distances between them. The case of linear products is considered separately, and sufficient conditions for attainability of the estimates are given for it. Examples of calculating the Gromov-Hausdorff distance between flat tori are given. It is proved that for any metric space $X$ of density $d(X)$, the Gromov-Hausdorff distance between it and its $l^\infty$-product (in which the number of factors corresponds to $d(X)$) is equal to half its diameter.

The Gromov-Hausdorff distance between $l^p$-products of metric spaces

Abstract

This paper studies -products of metric spaces and provides estimates for the Gromov-Hausdorff distances between them. The case of linear products is considered separately, and sufficient conditions for attainability of the estimates are given for it. Examples of calculating the Gromov-Hausdorff distance between flat tori are given. It is proved that for any metric space of density , the Gromov-Hausdorff distance between it and its -product (in which the number of factors corresponds to ) is equal to half its diameter.
Paper Structure (8 sections, 25 theorems, 70 equations)

This paper contains 8 sections, 25 theorems, 70 equations.

Table of Contents

  1. Basic Definitions and Preliminary Results
  2. General Estimates for Distances Between Products
  3. Remarks on the theorem: the case of continuous Gromov--Hausdorff distance
  4. Distance Estimation for Linear Products
  5. Several Lemmas
  6. Main Theorems
  7. Attainability of the Upper Bound for $l^{\infty}$-Products
  8. Gromov--Hausdorff Topological Distance

Key Result

Proposition 1.1

For any $X, Y \in \mathcal{GH}$ and arbitrary mappings $f \colon X \to Y$ and $g \colon Y \to X$, we have

Theorems & Definitions (61)

  • Proposition 1.1: BogatyTuzhilin
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Example 2.4
  • ...and 51 more