# Selection Lemmas for Various Geometric Objects

@article{Ashok2016SelectionLF,
title={Selection Lemmas for Various Geometric Objects},
author={P. Ashok and Sathish Govindarajan and Ninad Rajgopal},
journal={ArXiv},
year={2016},
volume={abs/1401.0443}
}
• Published 2016
• Mathematics, Computer Science
• ArXiv
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 P. This question has been widely explored for… Expand
2 Citations

#### Figures, Tables, and Topics from this paper

Matching points with disks with a common intersection
• Mathematics, Computer Science
• Discret. Math.
• 2019
It is proved that for any R and B such that | R | = | B | , there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. Expand
Upper bounds for stabbing simplices by a line
• Mathematics, Computer Science
• ArXiv
• 2020
This paper tries to determine the upper bounds yielded by two point sets, known as the " stretched grid" and the "stretched diagonal", and uses analytical and numerical software methods to do so. Expand

#### References

SHOWING 1-10 OF 24 REFERENCES
Hitting and Piercing Rectangles Induced by a Point Set
• Mathematics, Computer Science
• COCOON
• 2013
This work proves an exact result on the strong and the weak variant of the First Selection Lemma for rectangles and shows exact combinatorial bounds on the hitting set problem for two special classes of induced axis-parallel rectangles. Expand
On k-sets and applications
In this thesis, we study the notion of k-sets from discrete geometry and its applications to other mathematical problems. We prove that the number of k-sets of the set N0 of nonnegative latticeExpand
Hitting Simplices with Points in ℝ3
• Mathematics, Computer Science
• Discret. Comput. Geom.
• 2010
Improved bounds on the first selection lemma in ℝ3 are presented and it is proved that c3≤1/44≈0.00227, improving the previous best result of c3=0.000227 and making progress, for the three-dimensional case, on the open problems. Expand
Selecting Points that are Heavily Covered by Pseudo-Circles, Spheres or Rectangles
• Computer Science, Mathematics
• Combinatorics, Probability and Computing
• 2004
Several point selection theorems concerning objects ‘spanned’ by a finite set of points are proved and improve and generalize results of Chazelle, Edelsbrunner, Guibas, Hershberger, Seidel and Sharir. Expand
Lectures on discrete geometry
• J. Matousek
• Computer Science, Mathematics
• 2002
This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area. Expand
A Point in Many Triangles
• B. Bukh
• Mathematics, Computer Science
• Electron. J. Comb.
• 2006
We give a simpler proof of the result of Boros and Furedi that for any finite set of points in the plane in general position there is a point lying in $2/9$ of all the triangles determined by theseExpand
Stabbing Simplices by Points and Flats
• Mathematics, Computer Science
• Discret. Comput. Geom.
• 2010
An equipartition result of independent interest is established (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in ℝd can be divided into 4d−2 equal parts by 2d−1 hyperplanes intersecting in a common ( d−2)-flat. Expand
Selecting Heavily Covered Points
• Mathematics, Computer Science
• SIAM J. Comput.
• 1994
A collection of geometric selection lemmas is proved, such as the following: For any set $P$ of $n$ points in three-dimensional space and any set ${\cal S}$ of $m$ spheres, where each sphere passesExpand
Small strong epsilon nets
• Mathematics, Computer Science
• CCCG
• 2010
It is proved that a strong centerpoint exists for axis-parallel boxes in R^d and given exact bounds and this is extended to small strong @e-nets in the plane. Expand
Point Selections and Weak e-Nets for Convex Hulls
• Computer Science, Mathematics
• Comb. Probab. Comput.
• 1992
One of our results: Let X be a finite set on the plane, 0 < e < 1. Then there exists a set F (a weak e-net) of size at most 7/e such that every convex set containing at least e|X| elements of XExpand