The Katchalski-Lewis transversal problem for regular polygons
type: Item
instance of: scholarly article
title: The Katchalski-Lewis transversal problem for regular polygons
zbMATH Open document ID: 1413.52009
author: D. Kharzeev
published in: Acta Mathematica Hungarica
publication date: 2018-07-23T00:00:00Z
review text: Let \(\mathcal F\) be a family of plane convex bodies. We say that: (i) \(\mathcal F\) has property \(T(k)\) if \(| \mathcal F| \geq k\) and any of its subfamilies with \(k\) members has a line transversal; (ii) \(\mathcal F\) has property \(T-m\) if there exists a subfamily of \(\mathcal F\) with \(|\mathcal F \setminus \mathcal G| \leq m\) which has a line transversal. The main result of the paper is the following: Let \(\mathcal F\) be a finite \(T(3)\)-family of disjoint translates of a regular \(2n\)-gon, where \(n \geq 5\). The following statements hold: (i) \(\mathcal F\) has property \(T-3\) for \(5 \leq n \leq 34\); (ii) \(\mathcal F\) has property \(T-2\) for \(n \geq 34\).
reviewed by: Mircea Balaj
Mathematics Subject Classification ID: 52A35
zbMATH DE Number: 6909529
zbMATH Keywords: \(2n\)-gon
zbMATH Keywords: line transversal
zbMATH Keywords: translate
MaRDI profile type: Publication
full work available at URL: s10474-018-0823-x
OpenAlex ID: W2801193665
cites work: Q4328173
cites work: Über ein Problem aus der kombinatorischen Geometrie
cites work: On common transversals
cites work: New upper bound on the transversal width of \(T\)(3)-families of discs
cites work: Proof of the Katchalski-Lewis transversal conjecture for \(T(3)\)-families of congruent discs
cites work: New bounds on the Katchalski-Lewis transversal problem
cites work: Line transversals to unit disks
cites work: A conjecture of Grünbaum on common transversals.
cites work: Cutting Families of Convex Sets
cites work: Proof of Grünbaum's conjecture on common transversals for translates
Wikidata QID: Q129972862
DOI: 10.1007/S10474-018-0823-X
Recommended article: On \(k\)-convex polygons
Recommended article: Q4718724
Recommended article: On transverse triangulations
Recommended article: Regular polygons and transfinite diameter
Recommended article: The Minkowski problem for polytopes
Recommended article: The Katchalski-Lewis transversal problem in \(\mathbb R^{n}\)
Recommended article: Q3800652
Recommended article: A problem of Kusner on equilateral sets
Recommended article: Q4714893
Recommended article: Rigid polyboxes and Keller's conjecture
description: scientific article; zbMATH DE number 6909529