On sets of constant distance from a planar set

Abstract

In this paper we prove that $d$-boundaries
$$
D_d=\{x:\roman{dist}( x,Z) =d\}
$$
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.