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Proof of the Katchalski-Lewis Transversal Conjecture for T(3)-Families of Congruent Discs

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Abstract

A family of disjoint closed congruent discs is said to have property T(3) if to every triple of discs there exists a common line transversal. Katchalski and Lewis [10] proved the existence of a constant mdisc such that to every family of disjoint closed congruent discs with property T(3) a straight line can be found meeting all but at most mdisc of the members of the family. They conjectured that this is true even with mdisc = 2. On one hand Bezdek [1] proved mdisc ≥ 2 in 1991 and on the other hand Kaiser [9] showed mdisc ≤ 12 in a recent paper. The present work is devoted to proving this conjecture showing that mdisc ≤ 2.

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Correspondence to Aladar Heppes.

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Heppes, A. Proof of the Katchalski-Lewis Transversal Conjecture for T(3)-Families of Congruent Discs. Discrete Comput Geom 38, 289–304 (2007). https://doi.org/10.1007/s00454-007-1339-2

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Keywords

  • Unit Disc
  • Discrete Comput Geom
  • Support Line
  • Transversal Line
  • Transversal Width