Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets
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| Format: | Preprint |
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2024
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| _version_ | 1866917771820924928 |
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| author | David, Guy C. Romney, Matthew Schul, Raanan |
| author_facet | David, Guy C. Romney, Matthew Schul, Raanan |
| contents | A well-known open problem asks whether every bi-Lipschitz homeomorphism of $\mathbb{R}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into $\mathbb{R}^d$ factors into finitely many global bi-Lipschitz mappings of small distortion, outside of an exceptional set of arbitrarily small Lebesgue measure, which cannot in general be removed. Our main tool is a corona-type decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a related factorization result for bi-Lipschitz homeomorphisms of the $d$-sphere, and we show that bi-Lipschitz embeddings of the unit $d$-cube in $\mathbb{R}^d$ can be approximated by global piecewise affine homeomorphisms outside of a small set. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_05825 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets David, Guy C. Romney, Matthew Schul, Raanan Classical Analysis and ODEs 28A75, 57N45 A well-known open problem asks whether every bi-Lipschitz homeomorphism of $\mathbb{R}^d$ factors as a composition of mappings of small distortion. We show that every bi-Lipschitz embedding of the unit cube $[0,1]^d$ into $\mathbb{R}^d$ factors into finitely many global bi-Lipschitz mappings of small distortion, outside of an exceptional set of arbitrarily small Lebesgue measure, which cannot in general be removed. Our main tool is a corona-type decomposition theorem for bi-Lipschitz mappings. As corollaries, we obtain a related factorization result for bi-Lipschitz homeomorphisms of the $d$-sphere, and we show that bi-Lipschitz embeddings of the unit $d$-cube in $\mathbb{R}^d$ can be approximated by global piecewise affine homeomorphisms outside of a small set. |
| title | Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets |
| topic | Classical Analysis and ODEs 28A75, 57N45 |
| url | https://arxiv.org/abs/2409.05825 |