Abstract
We establish an embedding theorem for the Gromov–Hausdorff limit of a sequence of proper pointed metric spaces.
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Notes
Even though \({{{{\mathsf {dist}}_{\mathcal H}}}}\) is not always an actual distance function, we follow standard convention in calling it Hausdorff distance.
Alternatively, one can replace the distance in X with its associated standard bounded distance obtained by taking a minimum with 1, or one can do this with \({{{{\mathsf {dist}}_{\mathcal H}}}}\).
Even though \({{{{{\mathsf {dist}}}_{{\mathcal G \mathcal H}}}}}\) is not always an actual distance function, we follow standard convention in calling it Gromov–Hausdorff distance.
Even though \({{{{\mathsf {dist}}_{{{\mathcal G \mathcal H}}_*}}}}\) is not always an actual distance function, we follow standard convention in calling it pointed Gromov–Hausdorff distance.
Gromov (1981, p. 63) calls this “modified Hausdorff distance” and credits it to O. Gabber.
As there is no harm in doing so, we assume that each \(t_n\) satisfies \(t_n<1/2\).
Therefore, d is an \((\varepsilon ;a_n,b)\)-admissible distance on \(X_n\sqcup Z\), so \({{{{\mathsf {dist}}_{{{\mathcal G \mathcal H}}_*}}}}((X_n;a_n),(Z;b))\le \varepsilon \).
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D. A. Herron gratefully acknowledges partial support from the Charles Phelps Taft Research Center.
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Herron, D.A. Gromov–Hausdorff distance for pointed metric spaces. J Anal 24, 1–38 (2016). https://doi.org/10.1007/s41478-016-0001-x
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DOI: https://doi.org/10.1007/s41478-016-0001-x