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

We study the global topological structure and smoothness of the boundaries of $\epsilon$-neighbourhoods $E_{\epsilon }=\{x\in {\mathbb{R}}^2:\text{dist}(x,E)\le \epsilon \}$ of planar sets $E\subset {\mathbb{R}}^2$. We show that for a compact set $E$ and $\epsilon >0$ the boundary $\mathrm{\partial }{E}_{\epsilon }$ 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.