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

A. Blokh, M. Misiurewicz and L. Oversteegen

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 that then 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 ∂M or an arc A such that A∩∂M consists of both endpoints of A, and arcs and simple closed curves are dense in S(X,ε). In particular, if the boundary ∂M 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).