On sets of constant distance from a planar set

Abstract

In this paper we prove that $d$-boundaries $$ D_d=\{x:{\rm dist}( x,Z) =d\} $$ of a compact $Z \subset \mathbb{R}^{2}$ are closed absolutely continuous curves for $d$ greater than some constant depending on $Z$. It is also shown that $D_d$ is a trajectory of solution to the Cauchy Problem of a differential equation with a discontinuous right-hand side.