On Boundaries of $\varepsilon$-neighbourhoods of Planar Sets, Part II: Global Structure and Curvature

Abstract

We study the global topological structure and smoothness of the boundaries of $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of planar sets $E \subset \mathbb{R}^2$. 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. We also show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary.