Abstract
LetA be an arrangement ofn lines in the plane. IfR 1, …,R r arer distinct regions ofA, andR i is ap i-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.
Similar content being viewed by others
References
B. Grünbaum,Convex Polytopes. Wiley, London-New York-Sydney, 1967.
W. B. Carver,The polygonal regions into which a plane is divided by n straight lines, Amer. Math. Monthly48 (1941), 667–675.
F. Levi,Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade, Ber. math.-phys. Kl. sächs. Akad. Wiss. Leipzig78 (1926), 256–267.
Author information
Authors and Affiliations
Additional information
Financial support for this research was provided by the Carnegie Trust for the Universities of Scotland.
Rights and permissions
About this article
Cite this article
Canham, R.J. A theorem on arrangements of lines in the plane. Israel J. Math. 7, 393–397 (1969). https://doi.org/10.1007/BF02788872
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02788872