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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.06931 |
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| _version_ | 1866913262822490112 |
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| author | Honeycutt, Jacob Vellis, Vyron Zimmerman, Scott |
| author_facet | Honeycutt, Jacob Vellis, Vyron Zimmerman, Scott |
| contents | We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces $X$ so that any bi-Lipschitz embedding of a subset of the real line into $X$ extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset $Y$ of $X$ has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in $X$ by bi-Lipschitz curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06931 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bi-Lipschitz arcs in metric spaces with controlled geometry Honeycutt, Jacob Vellis, Vyron Zimmerman, Scott Metric Geometry We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supporting a Poincaré inequality). In particular, we find sharp conditions on metric measure spaces $X$ so that any bi-Lipschitz embedding of a subset of the real line into $X$ extends to a bi-Lipschitz embedding of the whole line. Along the way, we prove that if the complement of an open subset $Y$ of $X$ has small Assouad dimension, then it is a uniform domain. Finally, we prove a quantitative approximation of continua in $X$ by bi-Lipschitz curves. |
| title | Bi-Lipschitz arcs in metric spaces with controlled geometry |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2307.06931 |