The Number of Unit-Area Triangles in the Plane: Theme and Variation

Abstract

We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n20/9), improving the earlier bound O(n9/4) of Apfelbaum and Sharir [2]. We also show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans can be Ω(n2), for any triple of lines (it is always O(n2) in this case).

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Correspondence to Orit E. Raz.

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Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Orit E. Raz was also supported by a Shulamit Aloni Fellowship from the Israeli Ministry of Science. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. A preliminary version of this paper has appeared in Proc. 31st Annu. Sympos. Comput. Geom., 2015.

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Raz, O.E., Sharir, M. The Number of Unit-Area Triangles in the Plane: Theme and Variation. Combinatorica 37, 1221–1240 (2017). https://doi.org/10.1007/s00493-016-3440-8

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Mathematics Subject Classification (2000)

  • 52C10