A criterion for the equivalence of the Euclidean and the geodesic metric on compact subsets of $${\mathbb{R}^N}$$

Abstract

In (Studia Math 170:89–111, 2005), Bland and Feinstein show in two theorems that a compact plane set with pointwise (uniformly) regular boundary is itself pointwise (uniformly) regular. However, these statements do not cover simple sets as annular-like compact plane sets. In this note we prove that under suitable circumstances the statements keep true if one assumes the boundary to consist of finitely many pointwise (uniformly) regular sets.