2023/07/28 by Ting-Wei Chao, Chao, Ting-Wei, Hung-Hsun Hans Yu +1 · 1 citation
Computer Science · Mathematics · #05D40 #52C35 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2307.15380
openalex publication_date 2023/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A joint of a set of lines L in \mathbbFd is a point that is contained in d lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by L lines. Guth and Katz showed that the number of joints is at most O(L3/2) in ℝ3 using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the d-wise intersections and all the (d-1)-wise intersections of M hyperplanes in general position. Furthermore, this construction is conjectured to be optimal. In this paper, we verify the conjecture and show that this is the only optimal construction by using a more sophisticated polynomial method argument. This is the first tight bound and structural theorem obtained using this method. We also give a new definition of multiplicity that strengthens the main result of a previous work by Tidor, Zhao and the second author. Lastly, we relate the joints problem to some set-theoretic problems and prove conjectures of Bollobás and Eccles regarding partial shadows.