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

Abstract

We study the geometry, topological properties and smoothness of the boundaries of closed $\epsilon$-neighbourhoods $E_{\epsilon }=\{x\in {\mathbb{R}}^2:\text{dist}(x,E)\le \epsilon \}$ of compact planar sets $E\subset {\mathbb{R}}^2$. We develop a novel technique for analysing the boundary, and use this to obtain a classification of singularities (i.e.~non-smooth points) on $\partial E_{\epsilon }$ into eight categories. We show that the set of singularities is either countable or the disjoint union of a countable set and a closed, totally disconnected, nowhere dense set. Furthermore, we characterise, in terms of local geometry, those $\epsilon$-neighbourhoods whose complement $\overline{{\mathbb{R}}^2\setminus E_{\epsilon }}$ is a set with positive reach. It is known that for all bounded $E\subset {\mathbb{R}}^d$ and all $\epsilon >0$, the boundary $\partial E_{\epsilon }$ is $(d-1)$-rectifiable. Improving on this, we identify a sufficient condition for the boundary to be uniformly rectifiable, and provide an example of a planar $\epsilon$-neighbourhood that is not Ahlfors regular. In terms of the topological structure, 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. Finally, we show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary.

100 pages, 18 figures