On sets of constant distance from a planar set

Abstract

In this paper we prove that d-boundaries D d = {x : dist(x, Z) = d} of a compact Z ⊂ 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. 2000 Mathematics Subject Classification. 57N05, 34A36. Key words and phrases. d-boundary, absolutely continuous curve, differential equation with discontinuous right-hand side.

Old and new about equidistant sets and generalized conics

“…These sets have been studied widely in [2], [3] and recently in [7]. A deeper relation between midsets and ε−boundaries is indebted to Loveland:…”

“…We apply the above Lemma in order to characterize asymptotically the midset of two focal sets with a common supporting line. Notice that in the hypotheses of the next Proposition we drop conditions ( 6) and (7).…”

“…Proof. In order to apply Lemma 9, we need to show that we can recover conditions (6), (7). Since (−1, 0) is in A, for every h > 0 the foot points P (0,h) (A) belongs to the closed ball D h centered at (0, h) and passing through (−1, 0) (the same happens for P (0,h) (B), for the same ball D h since also passes trough (1, 0)).…”