Near-optimal generalisations of a theorem of Macbeath

Nabil H. Mustafa, Saurabh Ray

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The existence of Macbeath regions is a classical theorem in convex geometry ("A Theorem on nonhomogeneous lattices", Annals of Math, 1952). We refer the reader to the survey of I. Barany for several applications [3]. Recently there have been some striking applications of Macbeath regions in discrete and computational geometry. In this paper, we study Macbeath's problem in a more general setting, and not only for the Lebesgue measure as is the case in the classical theorem. We prove near-optimal generalizations for several basic geometric set systems. The problems and techniques used are closely linked to the study of ε-nets for geometric set systems.

Original languageEnglish (US)
Title of host publication31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages578-589
Number of pages12
Volume25
ISBN (Electronic)9783939897651
DOIs
StatePublished - Jan 1 2014
Event31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014 - Lyon, France
Duration: Mar 5 2014Mar 8 2014

Other

Other31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014
CountryFrance
CityLyon
Period3/5/143/8/14

Fingerprint

Computational geometry
Geometry

Keywords

  • Convex Geometry
  • Cuttings
  • Epsilon Nets
  • Geometric Set systems
  • Union Complexity

ASJC Scopus subject areas

  • Software

Cite this

Mustafa, N. H., & Ray, S. (2014). Near-optimal generalisations of a theorem of Macbeath. In 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014 (Vol. 25, pp. 578-589). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.STACS.2014.578

Near-optimal generalisations of a theorem of Macbeath. / Mustafa, Nabil H.; Ray, Saurabh.

31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014. Vol. 25 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2014. p. 578-589.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Mustafa, NH & Ray, S 2014, Near-optimal generalisations of a theorem of Macbeath. in 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014. vol. 25, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 578-589, 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014, Lyon, France, 3/5/14. https://doi.org/10.4230/LIPIcs.STACS.2014.578
Mustafa NH, Ray S. Near-optimal generalisations of a theorem of Macbeath. In 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014. Vol. 25. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2014. p. 578-589 https://doi.org/10.4230/LIPIcs.STACS.2014.578
Mustafa, Nabil H. ; Ray, Saurabh. / Near-optimal generalisations of a theorem of Macbeath. 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014. Vol. 25 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2014. pp. 578-589
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