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

TL;DR

This paper investigates the global topological and smoothness properties of boundaries of epsilon-neighborhoods of planar sets, revealing their structure as unions of Jordan curves and singularities, with curvature defined almost everywhere.

Contribution

It characterizes the boundary structure of epsilon-neighborhoods of planar sets, showing they are unions of Jordan curves and singularities, and establishes almost everywhere curvature on these curves.

Findings

Boundaries are unions of countably many Jordan curves and uncountable singularities.

Curvature is well-defined almost everywhere on the Jordan curve parts.

The structure provides insights into the geometric and topological properties of epsilon-neighborhoods.

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.