Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914095027978240 |
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| author | Li, Sean Schul, Raanan |
| author_facet | Li, Sean Schul, Raanan |
| contents | We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset $F$ and $f:F\to Y$, a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset $A\subset F$ so that $f$ is biLipschitz on $A$. We also give a characterization in terms of a full biLipschitz decomposition. These characterizations are new even for subsets of Euclidean space.
One of our tools is Alberti representations. On the way we give a method for constructing independent Alberti representations, which may be of independent interest. We use this to characterize unrectifiable metric spaces as those spaces for which there exist a positive measure subset $S$ and a Lipschitz map $ϕ$ into a lower dimensional Euclidean space so that $S$ is $\cH^1$-null with respect to all curve fragments that are quantitatively transversal to $ϕ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13525 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space Li, Sean Schul, Raanan Metric Geometry Classical Analysis and ODEs We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset $F$ and $f:F\to Y$, a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset $A\subset F$ so that $f$ is biLipschitz on $A$. We also give a characterization in terms of a full biLipschitz decomposition. These characterizations are new even for subsets of Euclidean space. One of our tools is Alberti representations. On the way we give a method for constructing independent Alberti representations, which may be of independent interest. We use this to characterize unrectifiable metric spaces as those spaces for which there exist a positive measure subset $S$ and a Lipschitz map $ϕ$ into a lower dimensional Euclidean space so that $S$ is $\cH^1$-null with respect to all curve fragments that are quantitatively transversal to $ϕ$. |
| title | Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space |
| topic | Metric Geometry Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2510.13525 |