How to take short cuts (Q1199128)

Let \(S\) be a rectifiable curve in Euclidean plane. The authors describe a geometric construction for adding short cuts so that every two points of \(S\) can be linked by a path along the union of \(S\) and the added short cuts. The construction guarantees that the length of the path is always at most the Euclidean distance of the linked points times a constant depending only on \(S\). The construction is provided for a polygon \(P\) approximating \(S\) and is based on the Voronoi polygons of the edges of \(P\). This approach supplies the not necessarily computable construction given by \textit{P. Jones} [Invent. Math. 102, No. 1, 1-15 (1990; Zbl 0731.30018)].