Selection Lemmas for Various Geometric Objects

Abstract

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set. In the first selection lemma, we consider the set of all the objects induced by a point set [Formula: see text]. This question has been widely explored for simplices in [Formula: see text], with tight bounds in [Formula: see text]. In our paper, we prove first selection lemma for other classes of geometric objects like boxes and balls in [Formula: see text]. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from [Formula: see text]. We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles and disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and balls in [Formula: see text]. In the second selection lemma, we consider an arbitrary [Formula: see text] sized subset of the set of all objects induced by [Formula: see text]. We study this problem for axis-parallel rectangles and show that there exists a point in the plane that is contained in [Formula: see text] rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir22 when [Formula: see text] is almost quadratic.