On sets of constant distance from a planar set. (English) Zbl 1030.54017

Pikuta, Piotr

Topol. Methods Nonlinear Anal. 21, No. 2, 369-374 (2003).

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)

Keywords:

planar set; \(d\)-boundaries; curves