On inducing polygons and related problems (Q359751)

It is totally settled in the affirmative a conjecture stated, and partially solved by \textit{P. Bose} et al. [Int. J. Comput. Geom. Appl. 13, No. 6, 447--462 (2003; Zbl 1062.68084)] regarding whether for every simple arrangement of \(n\) lines in general position in the plane there exists a simple \(n\)-gon \(P\) that induces the arrangement by extending every edge of \(P\) into a line. Moreover, it is also shown that such a polygon always exists and can be found in \(O(n \log n)\) time. It is left as an open question to show that this time complexity is the best possible.NEWLINENEWLINEIn a second part, the result is extended to curves in \({\mathbb R}^3\): For every finite family of curves \(C\) in general position it is shown that the union of the curves in \(C\) contains a simple cycle that visits every curve in \(C\) exactly once.