Gromov-Hausdorff Distances Between Spheres: Asymptotics and Vietoris-Rips Coindex Bounds
Abstract
For 1≤ m<n, let cm,n be the infimum of the scales r at which the Vietoris-Rips simplicial complex of the unit round sphere 𝕊m, equipped with its geodesic metric, admits a continuous odd map from 𝕊n. Previous work established the lower bound 1/2cm,n≤ dGH(𝕊m,𝕊n) through a quantitative Borsuk-Ulam theorem and raised the question of whether it is always sharp. We show that this bound can underestimate the distance by a factor arbitrarily close to two, the largest possible factor. Combining projective-code upper bounds for cm,n with lower bounds for dGH obtained from obstructions to homomorphisms between Borsuk graphs, we obtain explicit finite-dimensional bounds. Along sequences of pairs 1≤ m<n, we have cm,n→π/2 whenever log n/m→0, while dGH(𝕊m,𝕊n)→π/2 whenever m/n→0. When both conditions hold (for example when n=m2) the ratio of the distance to 1/2cm,n tends to two. For each fixed m≥3, the inequality is strict for all sufficiently large n.
Related
- Gromov-Hausdorff versus Hausdorff distances for Euclidean submanifolds with positive reach
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- Graph Distortion Distances, Gromov-Hausdorff and the Hardness of Gromov-Wasserstein
- Calibers of canonical hyperconnected spaces and their Vietoris hyperspaces
- Homotopy distributions for Vietoris-Rips complexes of random circle samples
BibTeX
@misc{lim2026gromovhausdorff,
title = {{Gromov-Hausdorff Distances Between Spheres: Asymptotics and Vietoris-Rips Coindex Bounds}},
author = {Sunhyuk Lim and Facundo Mémoli and Qingsong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.07778},
eprint = {2609.07778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07778},
}
RIS
TY - UNPB
TI - Gromov-Hausdorff Distances Between Spheres: Asymptotics and Vietoris-Rips Coindex Bounds
AU - Sunhyuk Lim
AU - Facundo Mémoli
AU - Qingsong Wang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2609.07778
UR - https://arxiv.org/abs/2609.07778
AB - For 1≤ m<n, let c_m,n be the infimum of the scales r at which the Vietoris-Rips simplicial complex of the unit round sphere 𝕊^m, equipped with its geodesic metric, admits a continuous odd map from 𝕊^n. Previous work established the lower bound 1/2c_m,n≤ d_GH(𝕊^m,𝕊^n) through a quantitative Borsuk-Ulam theorem and raised the question of whether it is always sharp. We show that this bound can underestimate the distance by a factor arbitrarily close to two, the largest possible factor. Combining projective-code upper bounds for c_m,n with lower bounds for d_GH obtained from obstructions to homomorphisms between Borsuk graphs, we obtain explicit finite-dimensional bounds. Along sequences of pairs 1≤ m<n, we have c_m,n→π/2 whenever log n/m→0, while d_GH(𝕊^m,𝕊^n)→π/2 whenever m/n→0. When both conditions hold (for example when n=m^2) the ratio of the distance to 1/2c_m,n tends to two. For each fixed m≥3, the inequality is strict for all sufficiently large n.
ER -