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
(spanned) by a point set P. This question has been widely explored for
simplices in Rd, with tight bounds in R2. In our paper,
we prove first selection lemma for other classes of geometric objects. We also
consider the strong variant of this problem where we add the constraint that
the piercing point comes from P. We prove an exact result on the strong and
the weak variant of the first selection lemma for axis-parallel rectangles,
special subclasses of axis-parallel rectangles like quadrants and slabs, disks
(for centrally symmetric point sets). We also show non-trivial bounds on the
first selection lemma for axis-parallel boxes and hyperspheres in
Rd.
In the second selection lemma, we consider an arbitrary m sized subset of
the set of all objects induced by P. We study this problem for axis-parallel
rectangles and show that there exists an point in the plane that is contained
in 24n4m3​ rectangles. This is an improvement over the previous
bound by Smorodinsky and Sharir when m is almost quadratic