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Gromov-Hausdorff distances between quotient metric spaces

Henry Adams, Armando Albornoz, Glenn Bruda, Jianda Du, Lodewyk Jansen van Rensburg, Alejandro Leon, Saketh Narayanan, Connor Panish, Chris Rugenstein, Martin Wall

Abstract

The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of spaces under group actions? Suppose a group $G$ acts by isometries on two metric spaces $X$ and $Y$. In this article, we study how the Hausdorff and Gromov-Hausdorff distances between $X$ and $Y$ and their quotient spaces $X/G$ and $Y/G$ are related. For the Hausdorff distance, we show that if $X$ and $Y$ are $G$-invariant subsets of a common metric space, then we have $d_{\mathrm{H}}(X,Y)=d_{\mathrm{H}}(X/G,Y/G)$. However, the Gromov-Hausdorff distance does not preserve this relationship: we show how to make the ratio $\frac{d_{\mathrm{GH}}(X/G,Y/G)}{d_{\mathrm{GH}}(X,Y)}$ both arbitrarily large and arbitrarily small, even if $X$ is an arbitrarily dense $G$-invariant subset of $Y$.

Gromov-Hausdorff distances between quotient metric spaces

Abstract

The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of spaces under group actions? Suppose a group acts by isometries on two metric spaces and . In this article, we study how the Hausdorff and Gromov-Hausdorff distances between and and their quotient spaces and are related. For the Hausdorff distance, we show that if and are -invariant subsets of a common metric space, then we have . However, the Gromov-Hausdorff distance does not preserve this relationship: we show how to make the ratio both arbitrarily large and arbitrarily small, even if is an arbitrarily dense -invariant subset of .

Paper Structure

This paper contains 13 sections, 7 theorems, 41 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Background and notation
  4. The Hausdorff distance under quotients
  5. The Gromov--Hausdorff distance under quotients
  6. Introductory examples
  7. The Gromov--Hausdorff distance for subsets under quotients
  8. Additional density assumptions
  9. Density alone is not enough
  10. A generalization in the arbitrarily dense setting
  11. Arbitrary density with arbitrary constants
  12. Addressing a question from HvsGH
  13. Conclusion and open questions

Key Result

Theorem 3.9

Given metric spaces $X$ and $Y$, the Gromov--Hausdorff distance between them can be equivalently defined as where the infimum is taken over all correspondences $R$ between $X$ and $Y$.

Figures (10)

  • Figure 1: The sets $X$ (in red) and $Y$ (in blue) are shown embedded in the plane on the left. On the right, they are embedded into a graph $G$.
  • Figure 2: Let $X$ be the set of red points, let $X^\varepsilon$ be the light red neighborhoods around them, and let $G$ be the group which reflects points across the x-axis. Then $X$ is $G$-invariant and so $X^\varepsilon$ must also be $G$-invariant.
  • Figure 3: A visual example for Theorem \ref{['thm:Hausdorff']}, where $G=\mathbb{Z}/4$ acts via rotation by multiples of $90^\circ$ around the origin.
  • Figure 4: (Left) The sets $X$ (in red) and $Y$ (in blue). (Right) The sets $X/G$ (in red) and $Y/G$ (in blue).
  • Figure 5: (Left) The sets $X$ (in red) and $Y$ (in blue). (right) The sets $X/G$ (in red) and $Y/G$ (in blue).
  • ...and 5 more figures

Theorems & Definitions (35)

  • Definition 3.1: Metric space
  • Definition 3.2: Epsilon-thickenings
  • Definition 3.3: Hausdorff distance
  • Definition 3.4
  • Definition 3.5: Gromov--Hausdorff distance
  • Example 3.6
  • Definition 3.7: Correspondence
  • Definition 3.8: Additive Distortion
  • Theorem 3.9: bridson2011metricBuragoBuragoIvanovkalton1999distances
  • Definition 3.10: Group Action
  • ...and 25 more