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The Gromov-Hausdorff distance between spheres

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arxiv 2105.00611 v6 pith:2KMFLISB submitted 2021-05-03 math.MG math.ATmath.DG

Sunhyuk Lim , Facundo Mémoli , Zane Smith This is my paper · ORCID
classification math.MGmath.ATmath.DG
keywords mathbbmathrmboundslowerdistancegromov-hausdorffinftyspheres
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abstract

We provide general upper and lower bounds for the Gromov-Hausdorff distance $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$ between spheres $\mathbb{S}^m$ and $\mathbb{S}^n$ (endowed with the round metric) for $0\leq m< n\leq \infty$. Some of these lower bounds are based on certain topological ideas related to the Borsuk-Ulam theorem. Via explicit constructions of (optimal) correspondences we prove that our lower bounds are tight in the cases of $d_{\mathrm{GH}}(\mathbb{S}^0,\mathbb{S}^n)$, $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^\infty)$, $d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^2)$, $d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^3)$ and $d_{\mathrm{GH}}(\mathbb{S}^2,\mathbb{S}^3)$. We also formulate a number of open questions.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line

    math.MG 2025-04 conditional novelty 7.0 of 10

    For unbounded A subset of R and bounded X, the curve A times_{l1} (tX) is a geodesic ray, and d_GH(Z^n, lambda Z^n) is at least 1/2 for every lambda > 1.

  2. Gromov-Hausdorff Geometry of Metric Trees

    math.MG 2024-12 conditional novelty 7.0 of 10

    For subsets of the real line and of infinite metric trees satisfying a boundary condition, the Gromov-Hausdorff distance to the ambient space equals the Hausdorff distance, and the canonical Hausdorff geodesic is a sh...

  3. Calculating Gromov-Hausdorff distance by means of asymptotic dimension

    math.MG 2025-05 conditional novelty 6.0 of 10

    The paper proves d_GH(Z^2,R^2)=sqrt(2)/2 and d_GH(A,R^2)=1/2 for a comb-shaped set A, using a new theorem that bounds GH distance below via r-disjoint covers and asymptotic dimension.

  4. Ultrametric spaces and clouds

    math.MG 2025-01 conditional novelty 5.0 of 10

    The ultrametrization map U is 1-Lipschitz on all metric spaces, preserves products with dotted connected spaces, and forces mutual exclusion of ultrametric and dotted connected spaces in unbounded clouds.

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