Disjoint empty convex polygons in planar point sets (Q5945676)

Let \(S\) be a set of \(n\) points in the plane, no three of which are collinear. A convex polygon whose vertices belong to \(S\) is an empty convex polygon if its interior is disjoint from \(S\). It is known that any set \(S\) admits \(\left[n\over 5\right]\) disjoint empty convex quadrangles. NEWLINENEWLINENEWLINEThe authors improve this lower bound by replacing \(\left[n\over 5\right]\) with \(\left[2n \over 9\right]\). This new bound is tight for \(n\leq 21\).