Discrete & Computational Geometry

, Volume 3, Issue 2, pp 97–102

Arrangements of lines with a minimum number of triangles are simple

  • Jean-Pierre Roudneff
Article

Abstract

Levi has shown that for every arrangement ofn lines in the real projective plane, there exist at leastn triangular faces, and Grünbaum has conjectured that equality can occur only for simple arrangements. In this note we prove this conjecture. The result does not hold for arrangements of pseudolines.

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References

  1. 1.
    B. Grünbaum,Arrangements and Spreads, Regional Conference Series in Mathematics 10, American Mathematical Society, Providence, RI, 1972.MATHGoogle Scholar
  2. 2.
    F. Levi, Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade,Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig 78 (1926), 256–267Google Scholar
  3. 3.
    J.-P. Roudneff, The maximum number of triangles in arrangements of (pseudo)-lines, submitted.Google Scholar
  4. 4.
    J.-P. Roudneff, Quadrilaterals and pentagons in arrangements of lines,Geom. Dedicata, to appear.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Jean-Pierre Roudneff
    • 1
  1. 1.Université P. et M. CurieParisFrance

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