$\epsilon$-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 $\partial E_{\epsilon }$ of its $\epsilon$-neighbourhood $E_{\epsilon }=\{x\in {\mathbb{R}}^2:\text{dist}(x,E)\le \epsilon \}$ 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 $\partial E_{\epsilon }$ 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 $\epsilon$-neighbourhood of one of their subsets.

10 pages, 3 figures; Added acknowledgments