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Acta Mathematica Hungarica

, Volume 155, Issue 1, pp 104–129 | Cite as

The Katchalski–Lewis transversal problem for regular polygons

  • Q. Du
  • L. Yuan
  • T. Zamfirescu
Article
  • 30 Downloads

Abstract

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).

Key words and phrases

2n-gon line transversal translate 

Mathematics Subject Classification

52A35 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHebei Normal UniversityShijiazhuangP.R. China
  2. 2.Hebei Key Laboratory of Computational Mathematics and ApplicationsShijiazhuangP.R. China
  3. 3.Fachbereich MathematikTechnische Universität DortmundDortmundGermany
  4. 4.Institute of Mathematics “Simion Stoilow”Roumanian AcademyBucharestRomania

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