$\varepsilon$-neighbourhoods in the Plane with a Nowhere-smooth Boundary
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912711966720000 |
|---|---|
| author | Lamb, Jeroen S. W. Rasmussen, Martin Timperi, Kalle G. |
| author_facet | Lamb, Jeroen S. W. Rasmussen, Martin Timperi, Kalle G. |
| contents | We give an example of a planar set $E\subset \mathbb{R}^2$ for which the boundary $\partial E_\varepsilon$ of its $\varepsilon$-neighbourhood $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ is nowhere $C^1$-smooth, in the sense that the set of singularities on the boundary is countably dense (where we note that the latter set cannot be uncountable). Furthermore, we give an example of a planar set $E$ for which $\partial E_\varepsilon$ has the same properties as above, but in addition contains an uncountable subset, with non-integer Hausdorff dimension, where curvature is not defined. Both constructions make use of a characterisation of those star-shaped sets that are an $\varepsilon$-neighbourhood of one of their subsets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09046 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\varepsilon$-neighbourhoods in the Plane with a Nowhere-smooth Boundary Lamb, Jeroen S. W. Rasmussen, Martin Timperi, Kalle G. Metric Geometry 51F30 (Primary) 57K20, 54C50, 58K40 (Secondary) We give an example of a planar set $E\subset \mathbb{R}^2$ for which the boundary $\partial E_\varepsilon$ of its $\varepsilon$-neighbourhood $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ is nowhere $C^1$-smooth, in the sense that the set of singularities on the boundary is countably dense (where we note that the latter set cannot be uncountable). Furthermore, we give an example of a planar set $E$ for which $\partial E_\varepsilon$ has the same properties as above, but in addition contains an uncountable subset, with non-integer Hausdorff dimension, where curvature is not defined. Both constructions make use of a characterisation of those star-shaped sets that are an $\varepsilon$-neighbourhood of one of their subsets. |
| title | $\varepsilon$-neighbourhoods in the Plane with a Nowhere-smooth Boundary |
| topic | Metric Geometry 51F30 (Primary) 57K20, 54C50, 58K40 (Secondary) |
| url | https://arxiv.org/abs/2511.09046 |