Incidences in three dimensions and distinct distances in the plane
Proceedings of the twenty-sixth annual symposium on Computational geometry
On lines, joints, and incidences in three dimensions
Journal of Combinatorial Theory Series A
The Joints Problem in $\mathbb{R}^n$
SIAM Journal on Discrete Mathematics
Computing the distance between piecewise-linear bivariate functions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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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.