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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.13554 |
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| _version_ | 1866915050568024064 |
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| author | Basso, Giuliano |
| author_facet | Basso, Giuliano |
| contents | In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows: If $X$ is any metric space and $A\subset X$ satisfies the condition $\text{Nagata}(n, c)$, then any $1$-Lipschitz map $f\colon A \to Y$ to a Banach space $Y$ admits a Lipschitz extension $F\colon X \to Y$ whose Lipschitz constant is at most $1000\cdot (c+1)\cdot \log_2(n+2)$. By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if $A\subset X$ consists of $n$ points, then Lipschitz extensions as above exist with a Lipschitz constant of at most $600 \cdot \log n \cdot (\log \log n)^{-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13554 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lipschitz extension theorems with explicit constants Basso, Giuliano Metric Geometry 54C20 (Primary) 54F45 (Secondary) In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows: If $X$ is any metric space and $A\subset X$ satisfies the condition $\text{Nagata}(n, c)$, then any $1$-Lipschitz map $f\colon A \to Y$ to a Banach space $Y$ admits a Lipschitz extension $F\colon X \to Y$ whose Lipschitz constant is at most $1000\cdot (c+1)\cdot \log_2(n+2)$. By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if $A\subset X$ consists of $n$ points, then Lipschitz extensions as above exist with a Lipschitz constant of at most $600 \cdot \log n \cdot (\log \log n)^{-1}$. |
| title | Lipschitz extension theorems with explicit constants |
| topic | Metric Geometry 54C20 (Primary) 54F45 (Secondary) |
| url | https://arxiv.org/abs/2310.13554 |