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Gromov--Hausdorff Distance Between Euclidean Unit Balls
Abstract
What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by dgh(Bm,Bn), for m>n? Note that the lower bound coming from the stability of persistent homology is zero, since all balls possess identical (trivial) persistent homology. To establish non-trivial lower bounds, we exploit the Borsuk--Ulam theorem. For any m > n ≥ 1, we prove that dgh(Bm,Bn)≥ 1/(1+αn) > 1/2, where we determine αn explicitly. We also prove that dgh(Bm,Bn)→ 1 as m→ ∞ and that dgh(Bm,Bn)<1 for all finite m>n≥ 1.
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BibTeX
@misc{adams2026gromovhausdorff,
title = {{Gromov--Hausdorff Distance Between Euclidean Unit Balls}},
author = {Henry Adams and Kushagri Sharma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09652},
eprint = {2609.09652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09652},
}
RIS
TY - UNPB
TI - Gromov--Hausdorff Distance Between Euclidean Unit Balls
AU - Henry Adams
AU - Kushagri Sharma
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.09652
UR - https://arxiv.org/abs/2609.09652
AB - What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by d_gh(B^m,B^n), for m>n? Note that the lower bound coming from the stability of persistent homology is zero, since all balls possess identical (trivial) persistent homology. To establish non-trivial lower bounds, we exploit the Borsuk--Ulam theorem. For any m > n ≥ 1, we prove that d_gh(B^m,B^n)≥ 1/(1+α_n) > 1/2, where we determine α_n explicitly. We also prove that d_gh(B^m,B^n)→ 1 as m→ ∞ and that d_gh(B^m,B^n)<1 for all finite m>n≥ 1.
ER -