Hitting Simplices with Points in ℝ3

Abdul Basit, Nabil H. Mustafa, Saurabh Ray, Sarfraz Raza

    Research output: Contribution to journalArticle

    Abstract

    The so-called first selection lemma states the following: given any set P of n points in ℝd, there exists a point in ℝd contained in at least cdnd+1-O(nd) simplices spanned by P, where the constant cd depends on d. We present improved bounds on the first selection lemma in ℝ3. In particular, we prove that c3≥0. 00227, improving the previous best result of c3≥0. 00162 by Wagner (On k-sets and applications. Ph. D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c3≤1/44≈0. 00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69-77, 1984) (where the two-dimensional case was settled).

    Original languageEnglish (US)
    Pages (from-to)637-644
    Number of pages8
    JournalDiscrete and Computational Geometry
    Volume44
    Issue number3
    DOIs
    StatePublished - May 28 2010

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    Lemma
    Triangle
    Open Problems
    Covering
    Three-dimensional

    Keywords

    • Centerpoint
    • Selection lemma
    • Simplex

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Geometry and Topology
    • Discrete Mathematics and Combinatorics
    • Computational Theory and Mathematics

    Cite this

    Hitting Simplices with Points in ℝ3 . / Basit, Abdul; Mustafa, Nabil H.; Ray, Saurabh; Raza, Sarfraz.

    In: Discrete and Computational Geometry, Vol. 44, No. 3, 28.05.2010, p. 637-644.

    Research output: Contribution to journalArticle

    Basit, A, Mustafa, NH, Ray, S & Raza, S 2010, 'Hitting Simplices with Points in ℝ3 ', Discrete and Computational Geometry, vol. 44, no. 3, pp. 637-644. https://doi.org/10.1007/s00454-010-9263-2
    Basit, Abdul ; Mustafa, Nabil H. ; Ray, Saurabh ; Raza, Sarfraz. / Hitting Simplices with Points in ℝ3 In: Discrete and Computational Geometry. 2010 ; Vol. 44, No. 3. pp. 637-644.
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