Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space
Abstract
Let (ℳ,dGH) denote the Gromov-Hausdorff space. We prove that every nonempty compact metric space is isometrically embeddable into (ℳ,dGH), answering a question of Iliadis. The corresponding result was previously known for finite metric spaces and several larger classes. However, the general compact case remained open. More precisely, let D>0 and let K be a nonempty compact metric space with diameter at most D. On a fixed Cantor space, we associate a metric to each $1$-Lipschitz function from K to [0,D] so that the Gromov-Hausdorff distance between any two resulting spaces is exactly the uniform distance between the corresponding functions. The resulting spaces have diameter at most 76D. As a consequence, the subspace of ℳ represented by Cantor spaces is itself isometrically universal for compact metric spaces. We also construct finite approximations of these spaces, obtain an exact formula for the Gromov-Hausdorff distance between the finite models, and derive uniform approximation and covering estimates.
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BibTeX
@misc{fukano2026every,
title = {{Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space}},
author = {Ryo Fukano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21469},
eprint = {2609.21469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21469},
}
RIS
TY - UNPB
TI - Every Compact Metric Space Is Isometrically Embeddable into the Gromov-Hausdorff Space
AU - Ryo Fukano
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21469
UR - https://arxiv.org/abs/2609.21469
AB - Let (ℳ,d_GH) denote the Gromov-Hausdorff space. We prove that every nonempty compact metric space is isometrically embeddable into (ℳ,d_GH), answering a question of Iliadis. The corresponding result was previously known for finite metric spaces and several larger classes. However, the general compact case remained open. More precisely, let D>0 and let K be a nonempty compact metric space with diameter at most D. On a fixed Cantor space, we associate a metric to each $1$-Lipschitz function from K to [0,D] so that the Gromov-Hausdorff distance between any two resulting spaces is exactly the uniform distance between the corresponding functions. The resulting spaces have diameter at most 76D. As a consequence, the subspace of ℳ represented by Cantor spaces is itself isometrically universal for compact metric spaces. We also construct finite approximations of these spaces, obtain an exact formula for the Gromov-Hausdorff distance between the finite models, and derive uniform approximation and covering estimates.
ER -