A criterion for the equivalence of the Euclidean and the geodesic metric on compact subsets of \({\mathbb{R}^N}\) (Q412021)

If \(K\) is a non-empty compact subset of \(\mathbb R^N\), then \(K\) is called rectifiably connected if \(K\) consists of more than one point and any two distinct points can be joined by a rectifiable path in \(K\). Let \(K\) be a rectifiably connected compact subset of \(\mathbb R^N\). Then \(K\) is said to be pointwise regular if for each \(z\in K\), there exists a constant \(c_z \geq 0\) (depending on \(z\) only) such that \(\delta_K (z,w) \leq c_z|z-w|\) holds for all \(w\in K\). \(K\) is said to be uniformly regular if there exists \(c \geq 0\) such that \(\delta_K (z,w) \leq c|z-w|\) holds for all \(z,w\in K\), where \(\delta K(x,y) = \inf\{L(\gamma) : \gamma\) is a continuous path in \(K\) joining \(x\) to \(y\}\), \(L(\gamma)\) being the length of \(\gamma\). The author proves that if \(\phi \neq\, L \subseteq \mathbb R^N\) is a compact set such that \(\partial L\) has only finitely many components \(K_1, \dots., K_n\) all of them being uniformly regular and such that each component of \(\mathbb R^N\setminus L\) has a connected boundary, then \(L\) itself is infinitely regular. This improves much the result of \textit{W. J. Bland} and \textit{J. F. Feinstein} [Stud. Math. 170, No. 1, 89--111 (2005; Zbl 1077.46017)].