Abstract Let L be a set of n lines in ℝ
d
, 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(n
d/(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.
d
, 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(n
d/(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.
- Content Type Journal Article
- DOI 10.1007/s00454-010-9246-3
- Authors
- Haim Kaplan, Tel Aviv University School of Computer Science Tel Aviv 69978 Israel
- Micha Sharir, Tel Aviv University School of Computer Science Tel Aviv 69978 Israel
- Eugenii Shustin, Tel Aviv University School of Mathematical Sciences Tel Aviv 69978 Israel
- Journal Discrete and Computational Geometry
- Online ISSN 1432-0444
- Print ISSN 0179-5376
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