On Lines and Joints

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Abstract

Let L be a set of n lines in ℝd, for d≥3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(nd/(d-1)). For d=3, this is a considerable simplification of the original algebraic proof of Guth and Katz (Algebraic methods in discrete analogs of the Kakeya problem, 4 December 2008, arXiv:0812.1043), and of the follow-up simpler proof of Elekes et al. (On lines, joints, and incidences in three dimensions. Manuscript, 11 May 2009, arXiv:0905.1583). Some extensions, e. g., to the case of joints of algebraic curves, are also presented.

Original languageEnglish
Pages (from-to)838-843
Number of pages6
JournalDiscrete and Computational Geometry
Volume44
Issue number4
DOIs
StatePublished - 2010

Keywords

  • Algebraic methods
  • Arrangements of lines in space
  • Joints
  • Kakeya problem

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