On Boundaries of ε-neighbourhoods of planar sets

Abstract

We study the geometric and topological structure of boundaries of ε-neighbourhoods of compact planar sets. The main contribution of the thesis is the development of a novel geometric framework for analysing the local properties of the boundaries of ε-neighbourhoods. We utilise the established concept of a tangent cone from set-valued analysis in our definition of outward directions, which allow us to investigate the tangential properties of the boundary by linking them to the characteristic geometry of ε-neighbourhoods. The technical foundation of the thesis is what we call the local contribution property, which intuitively asserts that the local geometry near a boundary point is completely determined by points that lie at most in two different directions relative to x. This result allows us to define local boundary representations, which provide a local description of the boundary. These are finite collections of graphs of Lipschitz-continuous functions, defined in local coordinate systems that are adapted to the orientations of so-called extremal outward directions. This tool turns out to provide a powerful technical framework for analysing also global properties of the boundary. Using these technical tools, we prove that the boundary points can be classified into smooth points and eight distinct categories of singularities. This classification permits a detailed analysis of the cardinalities and topological properties of the different types of singularities. We provide examples that show that so-called wedge singularities can lie densely on boundary segments that have positive one-dimensional Hausdorff measure. One of our main results is that the set of so-called chain singularities is closed and totally disconnected, which implies that it is nowhere dense on the boundary. Leveraging this analysis, we show that for a compact set E and ε > 0, the boundary is the disjoint union of a possibly uncountable set I of inaccessible singularities and an at most countably infinite collection J of Jordan curves. Finally, we prove the existence of curvature almost everywhere on the Jordan curve subsets of the boundary by showing that the derivatives of the functions constituting the local boundary representations are of bounded variation, and thus have a well-defined derivative at almost every point. Our work extends the existing literature by providing a complete classification of possible boundary geometries that applies generally for any compact set E and ε > 0. As opposed to the existing work on the topic, we approach the properties of the boundary from a geometric viewpoint, motivated by the need to understand the geometric and dynamical properties of ε-neighbourhoods appearing in set-valued dynamical systems as models for random dynamical systems with bounded noise.