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On \(p\)-Metric Spaces and the \(p\)-Gromov-Hausdorff Distance

Abstract

For each given \(p\in[1,\infty]\) we investigate certain sub-family \(\mathcal{M}_p\) of the collection of all compact metric spaces \(\mathcal{M}\) which are characterized by the satisfaction of a strengthened form of the triangle inequality which encompasses, for example, the strong triangle inequality satisfied by ultrametric spaces. We identify a one parameter family of Gromov-Hausdorff like distances \(\{d_{\mathrm{GH}}^{\scriptscriptstyle{(p)}}\}_{p\in[1,\infty]}\) on \(\mathcal{M}_p\) and study geometric and topological properties of these distances as well as the stability of certain canonical projections \(\mathfrak{S}_p:\mathcal{M}\rightarrow \mathcal{M}_p\). For the collection \(\mathcal{U}\) of all compact ultrametric spaces, which corresponds to the case \(p=\infty\) of the family \(\mathcal{M}_p\), we explore a one parameter family of interleaving-type distances and reveal their relationship with \(\{d_{\mathrm{GH}}^{\scriptscriptstyle{(p)}}\}_{p\in[1,\infty]}\).

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Notes

  1. Sometimes, we abbreviate \([x]_t^\theta\) to \([x]_t\) when the underlying dendrogram is clear from the context.

  2. If \((X,d_X)\) is moreover an ultrametric space, then \(\sim_0\) is exactly the same as the \(0\)-closed equivalence relation \(\sim_{\mathfrak{c}\left(0\right)}\) introduced in Section 2.2.3.

  3. Although [7, Proposition 26] states it only for finite cases, the same proof works for compact spaces.

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Acknowledgments

We thank Prof. Phillip Bowers from FSU for posing questions leading to the results in Chapter 2. We also thank Samir Chowdhury for interesting conversations about geodesics on Gromov-Hausdorff space. We thank Zane Smith who suggested studying the notion of kernel of the projection maps \(\mathfrak{S}_p\) which we discussed in Chapter 2.

Funding

This work was partially supported by the NSF through grants DMS-1723003, CCF-1740761, and CCF-1526513.

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Correspondence to Facundo Mémoli or Zhengchao Wan.

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Mémoli, F., Wan, Z. On \(p\)-Metric Spaces and the \(p\)-Gromov-Hausdorff Distance. P-Adic Num Ultrametr Anal Appl 14, 173–223 (2022). https://doi.org/10.1134/S2070046622030013

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Keywords

  • Gromov-Hausdorff distance
  • \(p\)-metric space
  • ultrametric space
  • interleaving distance