Ordinary lines in space

2018/03/26 by Frank de Zeeuw, de Zeeuw, Frank
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1803.09524

openalex publication_date 2018/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if a finite point set in real space does not have too many points on a plane, then it spans a quadratic number of ordinary lines. This answers the real case of a question of Basit, Dvir, Saraf, and Wolf. It shows that there is a significant difference in terms of ordinary lines between planar point sets, which may span a linear number of ordinary lines, and truly three-dimensional point sets. Our proof uses a projection argument of Kelly combined with a theorem of Beck on the number of spanned lines of a planar point set.

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