Work

Piercing intersecting convex sets

Year: 2025

Type: article

Abstract: 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.

Source: Linear Algebra and its Applications

DOI: 10.1016/j.laa.2025.02.007

Authors: Imre Bárány, Travis Dillon, Dömötör Pálvölgyi, Dániel Varga

Institutions: HUN-REN Alfréd Rényi Institute of Mathematics, University College London, Massachusetts Institute of Technology, Eötvös Loránd University

Language: English

Topic: Point processes and geometric inequalities

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,230