27 research outputs found
Stabbing simplices of point sets with k-flats
Let S be a set of n points inRdin general position.A set H of k-flats is called an mk-stabber of S if the relative interior of anym-simplex with vertices in S is intersected by at least one element of H. In thispaper we give lower and upper bounds on the size of mÃnimum mk-stabbers of point sets in Rd. We study mainly mk-stabbers in the plane and in R3Peer ReviewedPostprint (published version
Stabbing simplices of point sets with k-flats
Let S be a set of n points in Rd in general position. A set H of k-flats is called an mk-stabber of S if the relative interior of any m-simplex with vertices in S
is intersected by at least one element of H. In this paper we give lower and upper bounds on the size of minimum mk-stabbers of point sets in Rd. We study mainly mk-stabbers in the plane and in R3.Consejo Nacional de Ciencia y TecnologÃa (México)Ministerio de EconomÃa y CompetitividadGeneralitat de CatalunyaEuropean Science FoundationMinisterio de Ciencia e Innovació
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