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An Isometric Embedding of the $\ell^\infty$ product space of two bounded subspaces of the Gromov-Hausdorff Space into the Gromov-Hausdorff Space

Takuma Byakuno

TL;DR

The paper addresses embedding the ell-infinity product of bounded subspaces of the Gromov-Hausdorff space into the Gromov-Hausdorff space. It constructs gadget spaces S_{r,n}(X) encoding coordinate blocks and proves a distance equality that d_GH(S_{r,n}(X),S_{r,n}(Y)) equals the maximum coordinate distance, thereby establishing an isometric embedding of (M_{<=r})^n into B(5rn). The work yields corollaries for products of bounded subspaces and for bounded subspaces of M, advancing the understanding of universality and the structure of the GH space. Overall, the results provide explicit embedding techniques for coordinate-wise finite metric configurations within the Gromov-Hausdorff framework and contribute to the theory of metric space universality.

Abstract

In this paper, we prove the $\ell^\infty$ product space of two bounded subspaces of the Gromov-Hausdorff space can be isometrically embedded into the Gromov-Hausdorff space.

An Isometric Embedding of the $\ell^\infty$ product space of two bounded subspaces of the Gromov-Hausdorff Space into the Gromov-Hausdorff Space

TL;DR

The paper addresses embedding the ell-infinity product of bounded subspaces of the Gromov-Hausdorff space into the Gromov-Hausdorff space. It constructs gadget spaces S_{r,n}(X) encoding coordinate blocks and proves a distance equality that d_GH(S_{r,n}(X),S_{r,n}(Y)) equals the maximum coordinate distance, thereby establishing an isometric embedding of (M_{<=r})^n into B(5rn). The work yields corollaries for products of bounded subspaces and for bounded subspaces of M, advancing the understanding of universality and the structure of the GH space. Overall, the results provide explicit embedding techniques for coordinate-wise finite metric configurations within the Gromov-Hausdorff framework and contribute to the theory of metric space universality.

Abstract

In this paper, we prove the product space of two bounded subspaces of the Gromov-Hausdorff space can be isometrically embedded into the Gromov-Hausdorff space.

Paper Structure

This paper contains 4 sections, 4 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['theorem1']}
  4. Acknowledgments

Key Result

Proposition 1.2

All bounded metric subspaces of an $\ell^\infty$ normed space $\mathbb{R}^n$ can be isometrically embedded into the Gromov-Hausdorff space.

Figures (1)

  • Figure 1: The definition of $\mathop{\mathfrak{S}}\nolimits_{r,n}(X)$

Theorems & Definitions (7)

  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • proof : Proof of Theorem \ref{['theorem1']}
  • proof : Proof of Corollary \ref{['theorem2']}
  • proof : Proof of Corollary \ref{['theorem3']}