Piercing intersecting convex sets
Assume two finite families A and B of convex sets in R 3 have the property that A ∩ B ≠ ∅ for every A ∈ A and B ∈ B . Is there a constant γ > 0 (independent of A and B ) such that there is a line intersecting γ | A | sets in A or γ | B | sets in B ? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in A lie in parallel planes and all sets in B lie in parallel planes; in fact, one of the two families has a transversal by a single line.
Authors
- Imre Bárány (ORCID: https://orcid.org/0000-0001-7455-7467)
- Travis Dillon (ORCID: https://orcid.org/0000-0002-4250-8836)
- Dániel Varga (ORCID: https://orcid.org/0000-0002-8004-1704)
- Dömötör Pálvölgyi (ORCID: https://orcid.org/0000-0003-2970-0943)
Institutions
- Eötvös Loránd University (HU)
- HUN-REN Alfréd Rényi Institute of Mathematics (HU)
- University College London (GB)
- Massachusetts Institute of Technology (US)
Publication Details
- Journal
- Linear Algebra and its Applications
- Published
- 2025-02-05
- DOI
- https://doi.org/10.1016/j.laa.2025.02.007
- Primary Topic
- Computational Geometry and Mesh Generation
- Type
- article
- Field-Weighted Citation Impact
- 0.00