$\varepsilon$-neighbourhoods in the Plane with a Nowhere-smooth Boundary

TL;DR

This work constructs planar sets $E$ for which the boundary of the $\varepsilon$-neighbourhood, $\partial E_\varepsilon$, is as rugged as possible: in one case it consists only of wedges and shallow singularities with no $C^1$-smooth subcurve, and in another, curvature fails on an uncountable subset of the boundary that nonetheless has zero $1$-st Hausdorff measure. The authors develop a framework linking smoothness to metric projections and positive reach, and they realize the pathological boundary as the boundary of an $\varepsilon$-neighbourhood of a suitable star-shaped set via a two-step construction: a specially designed function on $[0,2\pi]$ wrapped around the origin by a polar map. A second construction augments this with a Cantor-function component to create an uncountable curvature-nonexistence set with positive Hausdorff dimension, all within the $\varepsilon$-neighbourhood paradigm. These results sharpen the understanding of the fine geometry of tubular neighbourhoods, with potential implications for the study of attractors and noisy dynamical systems where $\varepsilon$-neighbourhoods arise naturally.

Abstract

We give an example of a planar set $E\subset \mathbb{R}^2$ for which the boundary $\partial E_\varepsilon$ of its $\varepsilon$-neighbourhood $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ is nowhere $C^1$-smooth, in the sense that the set of singularities on the boundary is countably dense (where we note that the latter set cannot be uncountable). Furthermore, we give an example of a planar set $E$ for which $\partial E_\varepsilon$ has the same properties as above, but in addition contains an uncountable subset, with non-integer Hausdorff dimension, where curvature is not defined. Both constructions make use of a characterisation of those star-shaped sets that are an $\varepsilon$-neighbourhood of one of their subsets.

Figures (3)

• Figure 1: Possible local boundary geometries around an $\varepsilon$-neighbourhood boundary point $x \in \partial E_\varepsilon$ (red dot). At a smooth point (smooth) the boundary is locally a $C^1$-curve. At a wedge singularity (S1) the tangential directions at the boundary form an angle $\theta \in (0, \pi)$. One-sided (A4) and two-sided (S5) shallow singularities are accumulation points of wedge singularities, from one or two directions, respectively.
• Figure 2: (a) The integral $I$ (red dashed line) of the jump function $f$ (blue line) has a singularity at each rational $q \in \mathbb{Q} \cap (0, 2\pi]$. The smooth function $P$ with bounded curvature (yellow dashed line) satisfies $P(0) = I(2\pi) = L$. The sum function $S = P + I$ (dark green) satisfies $S'(0) = P'(0) + I'(0) = m_{n(0)} > 0 = S'(2\pi)$. (b) In polar coordinates $(x, r)$, the $\varepsilon$-neighbourhood boundary (dark green curve) is defined at each angle $x \in [0, 2\pi]$ by the radius $r(x) = P(x) + I(x)$. Wedges and shallow singularities are dense on the boundary. The black dots indicate the positions of the eighteen singularities with largest differences between the one-sided derivatives.
• Figure 3: Illustration of the functions defined in the proof of Theorem \ref{['Thm_non-existence_of_curvature']}.

Theorems & Definitions (15)

• Theorem 1: Existence of a nowhere-smooth boundary
• Definition 2.1: Smooth point, singularity
• Definition 2.2: Reach of a set
• Proposition 2.3: Characterisation of everywhere smooth boundary
• Theorem 2.4: Positive reach is preserved in bi-Lipschitz maps Federer_Curvature_measures
• Lemma 2.5
• proof
• Definition 2.6: Star-shaped set
• Proposition 2.7
• proof