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

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 $$n=12$$.

### MSC:

 52A37 Other problems of combinatorial convexity

### Keywords:

hidden vertices; polygonal paths

Zbl 0996.52006
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