TY - JOUR

T1 - On Lines and Joints

AU - Kaplan, Haim

AU - Sharir, Micha

AU - Shustin, Eugenii

N1 - Funding Information:
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.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Algebraic methods

KW - Arrangements of lines in space

KW - Joints

KW - Kakeya problem

UR - http://www.scopus.com/inward/record.url?scp=77957987458&partnerID=8YFLogxK

U2 - 10.1007/s00454-010-9246-3

DO - 10.1007/s00454-010-9246-3

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:77957987458

SN - 0179-5376

VL - 44

SP - 838

EP - 843

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -