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

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.