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Arrangements of lines with a minimum number of triangles are simple. (English) Zbl 0639.52007

The author proves the following conjecture of B. Grünbaum [Arrangements and spreads. Providence, R. I.: American Mathematical Society (AMS) (1972; Zbl 0249.50011)]: Any arrangement of \(n\) (straight) lines in the real projective plane defining exactly \(n\) triangular faces, the minimum possible number, is a simple arrangement. The proof runs by contradiction showing that any nonsimple arrangement of \(n\) (pseudo-)lines with \(n\) triangular faces necessarily contains a configuration which is not realizable by straight lines. In fact (as already known) the assertion is not true for arrangements of pseudolines.

MSC:

52A37 Other problems of combinatorial convexity
05B99 Designs and configurations

Citations:

Zbl 0249.50011
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References:

[1] B. Grünbaum,Arrangements and Spreads, Regional Conference Series in Mathematics 10, American Mathematical Society, Providence, RI, 1972. · Zbl 0249.50011
[2] F. Levi, Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade,Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig78 (1926), 256-267 · JFM 52.0575.01
[3] J.-P. Roudneff, The maximum number of triangles in arrangements of (pseudo)-lines, submitted. · Zbl 0837.51001
[4] J.-P. Roudneff, Quadrilaterals and pentagons in arrangements of lines,Geom. Dedicata, to appear. · Zbl 0617.51006
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