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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