Lower Bounds for Pinning Lines by Balls

2009/06/16 by Otfried Cheong, Cheong, Otfried, Xavier Goaoc +4
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #Point processes and geometric inequalities #cs.CG

paper · pdf · doi:10.48550/arxiv.0906.2924

arxiv created 2009/06/16 · openalex publication_date 2009/06/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A line L is a transversal to a family F of convex objects in R^d if it intersects every member of F. In this paper we show that for every integer d>2 there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the property that every subfamily of size 2d-2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957.

Related

• Line transversals to disjoint balls
• Line transversals in families of connected sets the plane
• Transversals to colorful intersecting convex sets
• Happy Ending or Many Concurrent Lines
• Depth with respect to a family of convex sets