On local Lipschitz one sets
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908984854708224 |
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| author | Wójcicki, Ziemowit M. |
| author_facet | Wójcicki, Ziemowit M. |
| contents | We study the local Lipschitz one subsets of a finite dimensional space, that is, sets for which there exists a continuous function whose local Lipschitz derivative is the characteristic function of said set. We give a characterization of a local Lipschitz one set on the real line in terms of a certain measure-theoretic density condition, which we call quasi-density. We show that any local Lipschitz one set needs to be quasi-dense, but the converse does not hold. Finally, we show that any regular closed subset of a normed space is a local Lipschitz one set, but there exist local Lipschitz one sets that are not regular closed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19704 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On local Lipschitz one sets Wójcicki, Ziemowit M. Functional Analysis 46T20, 26A16, 26A21 We study the local Lipschitz one subsets of a finite dimensional space, that is, sets for which there exists a continuous function whose local Lipschitz derivative is the characteristic function of said set. We give a characterization of a local Lipschitz one set on the real line in terms of a certain measure-theoretic density condition, which we call quasi-density. We show that any local Lipschitz one set needs to be quasi-dense, but the converse does not hold. Finally, we show that any regular closed subset of a normed space is a local Lipschitz one set, but there exist local Lipschitz one sets that are not regular closed. |
| title | On local Lipschitz one sets |
| topic | Functional Analysis 46T20, 26A16, 26A21 |
| url | https://arxiv.org/abs/2604.19704 |