Let $\Omega $ be a connected open set in the plane and $\gamma : [0,1] \to \overline {\Omega }$ a path such that $\gamma ((0,1)) \subset \Omega $ . We show that the path $\gamma $ can be “pulled tight” to a unique shortest path which is homotopic to $\gamma $ , via a homotopy h with endpoints fixed whose intermediate paths $h_t$ , for $t \in [0,1)$ , satisfy $h_t((0,1)) \subset \Omega $ . We prove this result even in the case when there is no path of finite Euclidean length homotopic to $\gamma $ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.

设 $\Omega $ 是平面上的连通开集， $\gamma : [0,1] \to \overline {\Omega }$ 是一条路径使得 $\gamma ((0,1)) \subset \Omega $ . 我们证明了路径 $\gamma $ 可以被“拉紧”到与 $\gamma $ 同伦的唯一最短路径，通过一个端点固定的同伦h其中间路径 $h_t$ ，对于 $t \in [0 ,1)$ ，满足 $h_t((0,1)) \subset \Omega $ 。即使在没有与 $\gamma $ 的有限欧几里得长度同伦的路径的情况下，我们也证明了这个结果在这样的同伦下。为此，我们提供了其他三个自然的、等效的“最短”路径概念。这项工作概括了先前具有简单闭合曲线边界的简单连接域的结果。