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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2602.12999 |
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| _version_ | 1866911446387916800 |
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| author | Borrelli, Vincent Follet, Jean-Baptiste Thibert, Boris |
| author_facet | Borrelli, Vincent Follet, Jean-Baptiste Thibert, Boris |
| contents | It is well-known since the seminal work of Herbert Federer [Trans. of the AMS, 1959] that submanifolds of class $C^{1,1}$ have positive reach. In this paper, we extend this property to less regular submanifolds by using the notion of $μ$-reach that was introduced in the 2000's. We first show that every compact $C^1$ submanifold of the Euclidean space $\E^n$ has positive $μ$-reach for all $μ<1$. We then show that intermediate regularities $C^{1,α}$ induce more quantitative results on the norm $\|\nabla \d_M\|$ of the generalized gradient of the distance function~$\d_M$ to the submanifold. More precisely, if $M\subset \E^n$ is a submanifold of class $C^{1,α}$, with $α<1$, then there exists a constant $C>0$ such that
$$\forall p\in\E^n\setminus M,\quad 1 - \| \nabla \d_M(p) \|^2 \leq C ~ \d_M(p)^{\frac{2 α}{1- α}}.$$ We finally show that the exponent $2α/(1-α)$ in this estimate is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12999 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Submanifolds of class $C^{1,α}$ and sets with positive $μ$-reach Borrelli, Vincent Follet, Jean-Baptiste Thibert, Boris Differential Geometry Metric Geometry It is well-known since the seminal work of Herbert Federer [Trans. of the AMS, 1959] that submanifolds of class $C^{1,1}$ have positive reach. In this paper, we extend this property to less regular submanifolds by using the notion of $μ$-reach that was introduced in the 2000's. We first show that every compact $C^1$ submanifold of the Euclidean space $\E^n$ has positive $μ$-reach for all $μ<1$. We then show that intermediate regularities $C^{1,α}$ induce more quantitative results on the norm $\|\nabla \d_M\|$ of the generalized gradient of the distance function~$\d_M$ to the submanifold. More precisely, if $M\subset \E^n$ is a submanifold of class $C^{1,α}$, with $α<1$, then there exists a constant $C>0$ such that $$\forall p\in\E^n\setminus M,\quad 1 - \| \nabla \d_M(p) \|^2 \leq C ~ \d_M(p)^{\frac{2 α}{1- α}}.$$ We finally show that the exponent $2α/(1-α)$ in this estimate is sharp. |
| title | Submanifolds of class $C^{1,α}$ and sets with positive $μ$-reach |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2602.12999 |