Barany, Imre;
Dillon, Travis;
Palvolgyi, Domotor;
Varga, Daniel;
(2025)
Piercing intersecting convex sets.
Linear Algebra and its Applications
, 710
pp. 405-417.
10.1016/j.laa.2025.02.007.
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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 Mart´ınez, 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, all sets from one of the two families has a line transversal.
| Type: | Article |
|---|---|
| Title: | Piercing intersecting convex sets |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.laa.2025.02.007 |
| Publisher version: | https://doi.org/10.1016/j.laa.2025.02.007 |
| Language: | English |
| Additional information: | © 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/). |
| Keywords: | Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, Helly-type theorems, Line transversals, Linear programming |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10219591 |
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