An optimal extension of the centerpoint theorem

Nabil H. Mustafa, Saurabh Ray

Research output: Contribution to journalArticle

Abstract

We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 47n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.

Original languageEnglish (US)
Pages (from-to)505-510
Number of pages6
JournalComputational Geometry: Theory and Applications
Volume42
Issue number6-7
DOIs
StatePublished - Aug 1 2009

Fingerprint

Theorem
Convex Sets
Hitting Set
Hits

Keywords

  • Centerpoint theorem
  • Combinatorial geometry
  • Discrete geometry
  • Extremal methods
  • Hitting convex sets
  • Weak -nets

ASJC Scopus subject areas

  • Geometry and Topology
  • Computer Science Applications
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

Cite this

An optimal extension of the centerpoint theorem. / Mustafa, Nabil H.; Ray, Saurabh.

In: Computational Geometry: Theory and Applications, Vol. 42, No. 6-7, 01.08.2009, p. 505-510.

Research output: Contribution to journalArticle

@article{b5f24aae678e4031b857cc3838ca39f3,
title = "An optimal extension of the centerpoint theorem",
abstract = "We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 47n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.",
keywords = "Centerpoint theorem, Combinatorial geometry, Discrete geometry, Extremal methods, Hitting convex sets, Weak -nets",
author = "Mustafa, {Nabil H.} and Saurabh Ray",
year = "2009",
month = "8",
day = "1",
doi = "10.1016/j.comgeo.2007.10.004",
language = "English (US)",
volume = "42",
pages = "505--510",
journal = "Computational Geometry: Theory and Applications",
issn = "0925-7721",
publisher = "Elsevier",
number = "6-7",

}

TY - JOUR

T1 - An optimal extension of the centerpoint theorem

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

PY - 2009/8/1

Y1 - 2009/8/1

N2 - We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 47n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.

AB - We prove an optimal extension of the centerpoint theorem: given a set P of n points in the plane, there exist two points (not necessarily among input points) that hit all convex sets containing more than 47n points of P. We further prove that this bound is tight. We get this bound as part of a more general procedure for finding small number of points hitting convex sets over P, yielding several improvements over previous results.

KW - Centerpoint theorem

KW - Combinatorial geometry

KW - Discrete geometry

KW - Extremal methods

KW - Hitting convex sets

KW - Weak -nets

UR - http://www.scopus.com/inward/record.url?scp=84867956198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867956198&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2007.10.004

DO - 10.1016/j.comgeo.2007.10.004

M3 - Article

AN - SCOPUS:84867956198

VL - 42

SP - 505

EP - 510

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6-7

ER -