On Boundaries of $\varepsilon$-neighbourhoods of Planar Sets: Singularities, Global Structure, and Curvature
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| Format: | Preprint |
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2020
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| _version_ | 1866908381655072768 |
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| author | Lamb, Jeroen S. W. Rasmussen, Martin Timperi, Kalle G. |
| author_facet | Lamb, Jeroen S. W. Rasmussen, Martin Timperi, Kalle G. |
| contents | We study the geometry, topological properties and smoothness of the boundaries of closed $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of compact planar sets $E \subset \mathbb{R}^2$. We develop a novel technique for analysing the boundary, and use this to obtain a classification of singularities (i.e.~non-smooth points) on $\partial E_\varepsilon$ into eight categories. We show that the set of singularities is either countable or the disjoint union of a countable set and a closed, totally disconnected, nowhere dense set. Furthermore, we characterise, in terms of local geometry, those $\varepsilon$-neighbourhoods whose complement $\overline{\mathbb{R}^2 \setminus E_\varepsilon}$ is a set with positive reach. It is known that for all bounded $E \subset \mathbb{R}^d$ and all $\varepsilon > 0$, the boundary $\partial E_\varepsilon$ is $(d-1)$-rectifiable. Improving on this, we identify a sufficient condition for the boundary to be uniformly rectifiable, and provide an example of a planar $\varepsilon$-neighbourhood that is not Ahlfors regular. In terms of the topological structure, we show that for a compact set $E$ and $\varepsilon > 0$ the boundary $\partial E_\varepsilon$ can be expressed as a disjoint union of an at most countably infinite union of Jordan curves and a possibly uncountable, totally disconnected set of singularities. Finally, we show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_13515 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On Boundaries of $\varepsilon$-neighbourhoods of Planar Sets: Singularities, Global Structure, and Curvature Lamb, Jeroen S. W. Rasmussen, Martin Timperi, Kalle G. Metric Geometry General Topology 51F30 (Primary) 57K20, 54C50, 58K40 (Secondary) We study the geometry, topological properties and smoothness of the boundaries of closed $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of compact planar sets $E \subset \mathbb{R}^2$. We develop a novel technique for analysing the boundary, and use this to obtain a classification of singularities (i.e.~non-smooth points) on $\partial E_\varepsilon$ into eight categories. We show that the set of singularities is either countable or the disjoint union of a countable set and a closed, totally disconnected, nowhere dense set. Furthermore, we characterise, in terms of local geometry, those $\varepsilon$-neighbourhoods whose complement $\overline{\mathbb{R}^2 \setminus E_\varepsilon}$ is a set with positive reach. It is known that for all bounded $E \subset \mathbb{R}^d$ and all $\varepsilon > 0$, the boundary $\partial E_\varepsilon$ is $(d-1)$-rectifiable. Improving on this, we identify a sufficient condition for the boundary to be uniformly rectifiable, and provide an example of a planar $\varepsilon$-neighbourhood that is not Ahlfors regular. In terms of the topological structure, we show that for a compact set $E$ and $\varepsilon > 0$ the boundary $\partial E_\varepsilon$ can be expressed as a disjoint union of an at most countably infinite union of Jordan curves and a possibly uncountable, totally disconnected set of singularities. Finally, we show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary. |
| title | On Boundaries of $\varepsilon$-neighbourhoods of Planar Sets: Singularities, Global Structure, and Curvature |
| topic | Metric Geometry General Topology 51F30 (Primary) 57K20, 54C50, 58K40 (Secondary) |
| url | https://arxiv.org/abs/2012.13515 |