22,559 research outputs found
Selection Lemmas for various geometric objects
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 . This question has been widely explored for
simplices in , with tight bounds in . 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 . 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
.
In the second selection lemma, we consider an arbitrary sized subset of
the set of all objects induced by . We study this problem for axis-parallel
rectangles and show that there exists an point in the plane that is contained
in rectangles. This is an improvement over the previous
bound by Smorodinsky and Sharir when is almost quadratic
Helly-Type Theorems in Property Testing
Helly's theorem is a fundamental result in discrete geometry, describing the
ways in which convex sets intersect with each other. If is a set of
points in , we say that is -clusterable if it can be
partitioned into clusters (subsets) such that each cluster can be contained
in a translated copy of a geometric object . In this paper, as an
application of Helly's theorem, by taking a constant size sample from , we
present a testing algorithm for -clustering, i.e., to distinguish
between two cases: when is -clusterable, and when it is
-far from being -clusterable. A set is -far
from being -clusterable if at least
points need to be removed from to make it -clusterable. We solve
this problem for and when is a symmetric convex object. For , we
solve a weaker version of this problem. Finally, as an application of our
testing result, in clustering with outliers, we show that one can find the
approximate clusters by querying a constant size sample, with high probability
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