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arXiv · 0906.0558

On lines and Joints

Haim Kaplan·Micha Sharir·Eugenii Shustin
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Abstract

Let $L$ be a set of $n$ lines in $\reals^d$, for $d\ge 3$. A {\em 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 orignal algebraic proof of Guth and Katz~\cite{GK}, and of the follow-up simpler proof of Elekes et al. \cite{EKS}.

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Haim Kaplan, Micha Sharir, Eugenii Shustin. 2009-06-02. On lines and Joints. https://arxiv.org/abs/0906.0558

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