Distortion from spheres into Euclidean spaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914518671556608 |
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| author | Dibble, James |
| author_facet | Dibble, James |
| contents | Any function from a round $n$-dimensional sphere of radius $r$ into $n$-dimensional Euclidean space must distort the metric additively by at least $\displaystyle \frac{πr}{1 + \sqrt{1 - \frac{2}{n+2}}}$ if $n$ is even and $\displaystyle \frac{πr}{1 + \sqrt{1 - \frac{2(n+2)}{(n+1)(n+3)}}}$ if $n$ is odd. This is proved using a fixed-point theorem of Granas that generalizes the classical theorem of Borsuk-Ulam to set-valued functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distortion from spheres into Euclidean spaces Dibble, James Metric Geometry General Topology 51F99, 54H25 (Primary) 52A40 (Secondary) Any function from a round $n$-dimensional sphere of radius $r$ into $n$-dimensional Euclidean space must distort the metric additively by at least $\displaystyle \frac{πr}{1 + \sqrt{1 - \frac{2}{n+2}}}$ if $n$ is even and $\displaystyle \frac{πr}{1 + \sqrt{1 - \frac{2(n+2)}{(n+1)(n+3)}}}$ if $n$ is odd. This is proved using a fixed-point theorem of Granas that generalizes the classical theorem of Borsuk-Ulam to set-valued functions. |
| title | Distortion from spheres into Euclidean spaces |
| topic | Metric Geometry General Topology 51F99, 54H25 (Primary) 52A40 (Secondary) |
| url | https://arxiv.org/abs/2504.02276 |