Lower bounds for the number of small convex \(k\)-holes (Q2444312)

Let \(S\) be a set of \(n\) points in the plane such that no three points of \(S\) lie on a common straight line and a simple polygon \(P\), spanned by \(k\)-points from \(S\), is called a \(k\)-hole of \(S\) if no other point of \(S\) is contained in the interior of \(P\). Denoting the least number of convex \(k\)-holes in \(S\) by \(h_k(n)\), the authors first obtain a better lower bound for \(h_5(n)\), especially for small values of \(n\) by fine tuning the proof given by \textit{O. Aichholzer} et al. [Lecture Notes in Computer Science 7579, 1--13 (2012; Zbl 1374.52020)]. Better lower bounds for the number of empty triangles and convex 4-holes which are in certain sense generated by convex 5-holes, are been obtained by using a recent result of \textit{A. García} [Lecture Notes in Computer Science 7579, 249--257 (2012; Zbl 1374.68663)].