2025/12/16 by Imre Bárány, Barany, Imre, Julia Q. D. Du +7
Computer Science · Mathematics · #51M04 #52C35 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2512.14518
openalex publication_date 2025/12/16 · openalex created_date 2025/12/18 · openalex updated_date 2026/07/28
The Sylvester-Gallai theorem states that for a finite set of points in the plane, if every line determined by any two of these points also contains a third, then the set is necessarily made of collinear points. In this paper, we first provide a counterexample in the plane when the point set is countably infinite but bounded. Then we consider a variant of the Sylvester-Gallai theorem where instead of a finite point set we have a finite family of convex sets in ℝd (d≥ 2). Finally, we present another variant of the Sylvester-Gallai theorem, when instead of point sets we have a finite family of line-segments in the plane.