### On sets of constant distance from a planar set

DOI: http://dx.doi.org/10.12775/TMNA.2003.022

#### 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.

$$

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.

#### Keywords

d-boundary; absolutely continuous curve; differential equation with discontinuous right-hand side

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