Simple closed curves contained in~$\varepsilon$-boundaries of planar sets

Abstract

The $\varepsilon$-boundary of a set ${A}\subseteq\mathbb{R}^2$ is the set $\{{p}\in\mathbb{R}^2:ρ({p},{A})=\varepsilon\}$, where $ρ$ is the Euclidean distance. We prove that if ${A},{B}\subseteq\mathbb{R}^2$ are nonempty, connected sets, ${A}$ is bounded, and $0<\varepsilon<ρ({A},{B})$, then the $\varepsilon$-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 $\varepsilon>0$ and ${A}\subseteq\mathbb{R}^2$ is a nonempty, bounded, connected set, then the boundary of each component of $\{{p}\in\mathbb{R}^2: ρ({p},{A})>\varepsilon\}$ is a simple closed curve. Another corollary of this theorem is that the $\varepsilon$-boundary of a nonempty, bounded, connected set ${A}\subseteq\mathbb{R}^2$ contains a simple closed curve bounding the domain that contains the open $\varepsilon$-neighbourhood of ${A}$. In all these statements the connectivity condition can be significantly weakened. We also show that, for all $\varepsilon>0$, the $\varepsilon$-boundary of a nonempty, bounded set ${A}\subseteq\mathbb{R}^2$ contains a simple closed curve.