On sets of constant distance from a planar set - Pikuta, Piotr (Topological Methods in Nonlinear Analysis, 2003)
In this paper we prove that $d$-boundaries$$D_d=\{x:\roman{dist}( x,Z) =d\} $$ of acompact $Z \subset \mathbb{R}^{2}$ are closed absolutely continuouscurves for $d$ greater than some constant depending on $Z$. It isalso shown that $D_d$ is a trajectory of solution to the CauchyProblem of a differential equation with a discontinuous right-handside.