Quantitative upper bounds on the Gromov-Hausdorff distance between spheres
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910575975464960 |
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| author | Harrison, Michael Jeffs, R. Amzi |
| author_facet | Harrison, Michael Jeffs, R. Amzi |
| contents | The Gromov-Hausdorff distance between two metric spaces measures how far the spaces are from being isometric. It has played an important and longstanding role in geometry and shape comparison. More recently, it has been discovered that the Gromov-Hausdorff distance between unit spheres equipped with the geodesic metric has important connections to Borsuk-Ulam theorems and Vietoris-Rips complexes.
We develop a discrete framework for obtaining upper bounds on the Gromov-Hausdorff distance between spheres, and provide the first quantitative bounds that apply to spheres of all possible pairs of dimensions. As a special case, we determine the exact Gromov-Hausdorff distance between the circle and any higher-dimensional sphere, and determine the precise asymptotic behavior of the distance from the 2-sphere to the $k$-sphere up to constants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_11237 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantitative upper bounds on the Gromov-Hausdorff distance between spheres Harrison, Michael Jeffs, R. Amzi Metric Geometry Algebraic Topology Combinatorics 51F30, 53C23, 52C17 The Gromov-Hausdorff distance between two metric spaces measures how far the spaces are from being isometric. It has played an important and longstanding role in geometry and shape comparison. More recently, it has been discovered that the Gromov-Hausdorff distance between unit spheres equipped with the geodesic metric has important connections to Borsuk-Ulam theorems and Vietoris-Rips complexes. We develop a discrete framework for obtaining upper bounds on the Gromov-Hausdorff distance between spheres, and provide the first quantitative bounds that apply to spheres of all possible pairs of dimensions. As a special case, we determine the exact Gromov-Hausdorff distance between the circle and any higher-dimensional sphere, and determine the precise asymptotic behavior of the distance from the 2-sphere to the $k$-sphere up to constants. |
| title | Quantitative upper bounds on the Gromov-Hausdorff distance between spheres |
| topic | Metric Geometry Algebraic Topology Combinatorics 51F30, 53C23, 52C17 |
| url | https://arxiv.org/abs/2309.11237 |