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

TL;DR

The paper provides a complete geometric and topological analysis of the boundaries ∂E_ε of closed ε-neighborhoods of compact planar sets E. It develops a local boundary representation to classify every boundary point into eight singularity types (S1–S8) and shows that ∂E_ε decomposes into an inaccessible singularity set plus a countable union of Jordan curves, with curvature defined almost everywhere on the Jordan-curve parts. It proves that the set of chain singularities is closed and totally disconnected, while wedges and several other singularities are at most countable, and it links these local geometries to global properties such as positive reach of the complement and uniform rectifiability under suitable conditions. The results extend to a global topological structure of the boundary, establish a sufficient condition for uniform rectifiability, provide a counterexample to Ahlfors regularity, and prove curvature existence via boundary graphs and BV arguments, thereby offering a comprehensive framework for the boundary geometry of ε-neighborhoods in the plane.

Abstract

We study the geometry, topological properties and smoothness of the boundaries of closed $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ 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 $\partial E_\varepsilon$ 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 $\varepsilon$-neighbourhoods whose complement $\overline{\mathbb{R}^2 \setminus E_\varepsilon}$ is a set with positive reach. It is known that for all bounded $E \subset \mathbb{R}^d$ and all $\varepsilon > 0$, the boundary $\partial E_\varepsilon$ 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 $\varepsilon$-neighbourhood that is not Ahlfors regular. In terms of the topological structure, 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. Finally, we show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary.

Figures (18)

• Figure 1: Schematic illustration of the types of singularities mentioned in Theorem \ref{['Thm_Main_1']}. The grey area represents the $\varepsilon$-neighbourhood $E_\varepsilon$ and the white area the complement $\mathbb{R}^2 \setminus E_\varepsilon$. Every boundary point $x \in \partial E_\varepsilon$ either is a smooth point or belongs to exactly one of eight categories of singularities. At a wedge (type S1) the one-sided tangents form an angle $0 < \theta < \pi$. A sharp singularity (type S2) and a sharp-sharp singularity (type S3) can be thought of as extremal cases of a wedge, with $\theta = 0$. A shallow singularity (type S4) and a shallow-shallow singularity (type S5) have a well-defined tangent, but they are accumulation points (from one or two directions, respectively) of sequences of increasingly obtuse wedges (black dots). A chain singularity (type S6), a chain-chain singularity (type S7) and a sharp-chain singularity (type S8) share the geometric property of being accumulation points of sequences of increasingly acute wedges (black dots). This turns out to be equivalent (see Proposition \ref{['Prop_Topological_Characterisation_of_Chain_Singularities']}) to the topological property of being the limit with respect to Hausdorff distance of a sequence of disjoint connected components of the complement $\mathbb{R}^2 \setminus E_\varepsilon$. See also Figure \ref{['Figure_Three_Basic_Cases']}.
• Figure 2: Schematic illustration of Theorem \ref{['Thm_global_structure']}. ( A): Jordan curve subsets of the boundary of a connected component $U$ of the complement $\mathbb{R}^2 \setminus E_\varepsilon$. Here $\partial U = J_1 \cup J_2$ and $J_1 \cap J_2 = \{x_1\}$. At the centre of the figure there is another connected component $V \subset \mathbb{R}^2 \setminus E_\varepsilon$, with $\partial V \cap \partial U = \{x_2, x_3\}$. Note that only the sharp-sharp singularities $x_1, x_2$ and $x_3$ present a choice regarding how to continue the boundary curves. ( B): The two types of inaccessible singularities. A chain singularity (type S6) is the limit (in terms of Hausdorff distance) of a sequence of mutually disjoint connected components of the complement. Such a sequence approaches a chain-chain singularity (type S7) on both sides. These are the only types of boundary points that do not lie on Jordan curve subsets on the boundary.
• Figure 3: Schematic illustration of the difference between the $\varepsilon$-boundary $\partial E_{<\varepsilon}$ and the boundary of the closed $\varepsilon$-neighbourhood $\partial E_\varepsilon$. For $E = E_1 \cup E_2$, the set $A := \partial E_{<\varepsilon} \setminus \partial E_\varepsilon$ is non-empty. See also the discussion following Lemma \ref{['Lemma_OD_versus_dot_product_ineq']} and Rataj_Winter_On_Volume.
• Figure 4: Illustration of the relationship between outward directions and extremal contributors. ( A) A singularity $x \in \partial E_\varepsilon$ with $\Pi_E^{\mathrm{ext}}(x) = \{y_1, y_2\}$ and $\Xi_{x}^{\mathrm{ext}}(E_\varepsilon) = \{\xi_1, \xi_2\}$. ( B) The set $\Xi_x(E_\varepsilon) \subset S^1$ of outward directions is geodesically convex with boundary $\partial_{S^1} \Xi_x(E_\varepsilon) = \{\xi_1, \xi_2\}$.
• Figure 5: Construction of the sequence $(z_n)_{n=1}^\infty$. The singularity $x$ is here depicted as a wedge, but the procedure is the same for other types of boundary points. ( A) For each $n \in \mathbb{N}$, the variables $h_n, r_n$ satisfy $r_n = \varepsilon - \sqrt{\varepsilon^2 - h_n^2}$ and $\left\lVert H_n\right\rVert = \left\lVert\varphi_n\right\rVert$. ( B) The point $z_n \in \partial E_\varepsilon$ lies on a geodesic arc-segment on $\partial B_{\left\lVert\varphi_n\right\rVert}(x)$ that connects the points $x_n$ and $x + H_n$.

Theorems & Definitions (114)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Definition 1.1: Critical points, critical values, and regular points of the distance function Fu_Tubular_neighborhoods
• Definition 1.2: The reach of a set Federer_Curvature_measures
• Definition 1.3: Rataj_Zajicek_Smallness