Shortest paths in simply connected regions in \({\mathbb{R}}^ 2\) (Q910004)

Let \({\mathcal R}\) denote a topological 2-disk in the Euclidean plane (Jordan region). The authors show that for any two points p, q of \({\mathcal R}\) there exists a unique shortest path M(p,q) in \({\mathcal R}\). In case there is no path of finite length between p and q `shortest' is defined by local properties of the path: If \(z\in M(p,q)\) is an interior point of \({\mathcal R}\) then M(p,q) is locally a segment of a straight line. If \(z\in M(p,q)\) lies on the boundary \(\partial {\mathcal R}\) of \({\mathcal R}\) then \(\partial {\mathcal R}\) satisfies at z a certain support property (existence of a dividing half disk). For the proofs no further assumption on the boundary of \({\mathcal R}\) is necessary.