Inner Lipschitz approximation in o-minimal structures
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910043830484992 |
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| author | Nguyen, Nhan Valette, Anna Valette, Guillaume |
| author_facet | Nguyen, Nhan Valette, Anna Valette, Guillaume |
| contents | Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure
admits $\mathscr{C}^\infty$ cell decomposition, we show that the approximation can be required to be $\mathscr{C}^\infty$ and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06443 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inner Lipschitz approximation in o-minimal structures Nguyen, Nhan Valette, Anna Valette, Guillaume Algebraic Geometry 03C64, 14P99, 26A16, 41A44, 57R12 Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure admits $\mathscr{C}^\infty$ cell decomposition, we show that the approximation can be required to be $\mathscr{C}^\infty$ and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems. |
| title | Inner Lipschitz approximation in o-minimal structures |
| topic | Algebraic Geometry 03C64, 14P99, 26A16, 41A44, 57R12 |
| url | https://arxiv.org/abs/2603.06443 |