Simple closed curves contained in ε-boundaries of planar sets
[EN] The ε-boundary of a set A ⊆ R2 is the set { p ∈ R2 : ρ(p,A) = ε } , where ρ is the Euclidean distance. We prove that if A,B ⊆ R2 are nonempty, connected sets, A is bounded, and 0< ε < ρ(A,B), then the ε-boundary of A contains a simple closed curve (aka a Jordan curve) that separat...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2025 |
| Country: | España |
| Institution: | Universitat Politècnica de València (UPV) |
| Repository: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Language: | English |
| OAI Identifier: | oai:dnet:riunet______::63b5a2b8170fa6705cda03ce40eea592 |
| Online Access: | https://riunet.upv.es/handle/10251/234645 |
| Access Level: | Open access |
| Keyword: | Simple closed curve Jordan curve ε-boundaries Level set Distant sphere |
| Summary: | [EN] The ε-boundary of a set A ⊆ R2 is the set { p ∈ R2 : ρ(p,A) = ε } , where ρ is the Euclidean distance. We prove that if A,B ⊆ R2 are nonempty, connected sets, A is bounded, and 0< ε < ρ(A,B), then the ε-boundary of A contains a simple closed curve (aka a Jordan curve) that separates A and B. This statement follows from the theorem which says that if ε>0 and A ⊆ R2 is a nonempty, bounded, connected set, then the boundary of each component of { p ∈ R2 : ρ(p,A) > ε } is a simple closed curve. Another corollary of this theorem is that the ε-boundary of a nonempty, bounded, connected set A ⊆ R2 contains a simple closed curve bounding the domain that contains the open ε-neighbourhood of A. In all these statements the connectivity condition can be significantly weakened. We also show that, for all ε>0, the ε-boundary of a nonempty, bounded set A ⊆ R2 contains a simple closed curve. |
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