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

Abstract

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 $\partial E_{\epsilon }$ 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.

This article has been superseded by arXiv:2012.13515