Patrakeev, MikhailVolkov, Aleksei2026-04-292026-04-292025-12-031576-9402https://riunet.upv.es/handle/10251/234645[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.Reconocimiento - No comercial - Compartir igual (by-nc-sa)Simple closed curveJordan curveε-boundariesLevel setDistant sphereArtículo10.4995/agt.24499Abierto1989-4147