Sets of constant distance from a compact set in 2-manifolds with a geodesic metric

Academic Article

Abstract

  • Let (M, d) be a complete topological 2-manifold, possibly with boundary, with a geodesic metric d.Let X ∪ M be a compact set. We show then that for all but countably many ε each component of the set S(X, ε)of points ε-distant from X is either a point, a simple closed curve disjoint from dM or an arc A such that A ∩M consists of both endpoints of A and that arcs and simple closed curves are dense in S(X, ε). In particular, if the boundary dM of M is empty, then each component of the set S(X,ε) is either a point or a simple closed curve and the simple closed curves are dense in S(X, ε). © 2008 American Mathematical Society.
  • Digital Object Identifier (doi)

    Author List

  • Blokh A; Misiurewicz M; Oversteegen L
  • Start Page

  • 733
  • End Page

  • 743
  • Volume

  • 137
  • Issue

  • 2