Planar Bilipschitz Extension from Separated Nets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914408962195456 |
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| author | Dymond, Michael Kaluža, Vojtěch |
| author_facet | Dymond, Michael Kaluža, Vojtěch |
| contents | We prove that every $L$-bilipschitz mapping $\mathbb{Z}^2\to\mathbb{R}^2$ can be extended to a $C(L)$-bilipschitz mapping $\mathbb{R}^2\to\mathbb{R}^2$ and provide a polynomial upper bound for $C(L)$. Moreover, we extend the result to every separated net in $\mathbb{R}^2$ instead of $\mathbb{Z}^2$, with the upper bound gaining a polynomial dependence on the separation and net constants associated to the given separated net. This answers an Oberwolfach question of Navas from 2015 and is also a positive solution of the two-dimensional form of a decades old open (in all dimensions at least two) problem due to Alestalo, Trotsenko and Väisälä. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22294 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Planar Bilipschitz Extension from Separated Nets Dymond, Michael Kaluža, Vojtěch Metric Geometry Functional Analysis We prove that every $L$-bilipschitz mapping $\mathbb{Z}^2\to\mathbb{R}^2$ can be extended to a $C(L)$-bilipschitz mapping $\mathbb{R}^2\to\mathbb{R}^2$ and provide a polynomial upper bound for $C(L)$. Moreover, we extend the result to every separated net in $\mathbb{R}^2$ instead of $\mathbb{Z}^2$, with the upper bound gaining a polynomial dependence on the separation and net constants associated to the given separated net. This answers an Oberwolfach question of Navas from 2015 and is also a positive solution of the two-dimensional form of a decades old open (in all dimensions at least two) problem due to Alestalo, Trotsenko and Väisälä. |
| title | Planar Bilipschitz Extension from Separated Nets |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2410.22294 |