On sets of constant distance from a planar set. (English) Zbl 1030.54017
Summary: We prove that \(d\)-boundaries \(D_d=\{x: \text{dist} (x,Z)=d\}\) 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 a solution to the Cauchy problem of a differential equation with a discontinuous right-hand side.
MSC:
| 54E35 | Metric spaces, metrizability |
| 34A36 | Discontinuous ordinary differential equations |
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
