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(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.
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Work on this paper has been partly supported by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work by Micha Sharir was also supported by NSF Grants CCF-05-14079 and CCF-08-30272, by Grant 155/05 from the Israel Science Fund. and by Grant 2006/194 from the U.S.—Israeli Binational Science Foundation. Work by Haim Kaplan was also supported by Grant 975/06 from the Israel Science Fund, and by Grant 2006/204 from the U.S.—Israel Binational Science Foundation.
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Kaplan, H., Sharir, M. & Shustin, E. On Lines and Joints. Discrete Comput Geom 44, 838–843 (2010). https://doi.org/10.1007/s00454-010-9246-3
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DOI: https://doi.org/10.1007/s00454-010-9246-3