The lower bound of the number of non-overlapping triangles. (Q1871441)

If \(\mathcal P\) is a convex \(n\)-gon with sides \(s_1,s_2, \dots ,s_n\) and \({\mathcal A}=\{A_1,A_2, \dots ,A_n\}\) a point set of the plane then there exists a matching of the sides of \(\mathcal P\) and the points of \(\mathcal A\) such that among the triangles spanned by the matching points and sides there are at least \(\left\lceil \frac{n}{2} \right\rceil\) triangles which are mutually non-overlapping. This improves a result of \textit{A. Bezdek} [Geom. Dedicata 80, 73--79 (2000; Zbl 0981.52012)]. Moreover, the authors show that this lower bound is best possible.