The problem of polygons with hidden vertices. (English) Zbl 1054.52005

Ling, Joseph M.

Beitr. Algebra Geom. 45, No. 1, 217-224 (2004).

Let \(P=[P_{1}P_{2}\dots P_{n}]\) be a polygonal path in \({\mathbb R}^3\) without self intersection. Ewald asked the question whether there exists such a polygon together with some point \(M\) not on the polygon with the property that for any \(i\) there exists some \(j\) such that the line segments \([M,P_{i}]\) and \([P_{j-1},P_{j}]\) intersect at a point in \((P_{j-1},P_{j})\). In [Beitr. Algebra Geom. 42, No. 2, 439–442 (2001; Zbl 0996.52006)] G. Ewald gave an example of such a configuration with \(n=14\). Ewald asked for the smallest number of vertices \(n_{\min}\) for which such a configuration exists, and proved \(8 \leq n_{\min} \leq 14\).
In the paper under review the author improves this inequality to \(11 \leq n_{\min} \leq 12\). Moreover, the author gives an example for \(n=12\).

Reviewer: Richard Bödi (Rüschlikon)

MSC:

52A37

Other problems of combinatorial convexity

Keywords:

hidden vertices; polygonal paths

Citations:

Zbl 0996.52006