Israel Journal of Mathematics

, Volume 7, Issue 4, pp 393–397

A theorem on arrangements of lines in the plane

  • R. J. Canham
Article

Abstract

LetA be an arrangement ofn lines in the plane. IfR1, …,Rr arer distinct regions ofA, andRi is api-gon (i=1, …,r) then we show that\(\sum\limits_{i = 1}^r {P_i \leqq n + 4} \left( {_2^r } \right)\). Further we show that for allr this bound is the best possible ifn is sufficiently large.

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References

  1. 1.
    B. Grünbaum,Convex Polytopes. Wiley, London-New York-Sydney, 1967.MATHGoogle Scholar
  2. 2.
    W. B. Carver,The polygonal regions into which a plane is divided by n straight lines, Amer. Math. Monthly48 (1941), 667–675.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    F. Levi,Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade, Ber. math.-phys. Kl. sächs. Akad. Wiss. Leipzig78 (1926), 256–267.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1969

Authors and Affiliations

  • R. J. Canham
    • 1
  1. 1.Michigan State UniversityEast Lansing

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