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

Authors Orit E. Raz, Micha Sharir

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Orit E. Raz
Micha Sharir

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Orit E. Raz and Micha Sharir. The Number of Unit-Area Triangles in the Plane: Theme and Variations. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 569-583, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n^{20/9}), improving the earlier bound O(n^{9/4}) of Apfelbaum and Sharir. We also consider two special cases of this problem: (i) We 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 Omega(n^2), for any triple of lines (it is always O(n^2) in this case). (ii) We show that if S is a convex grid of the form A x B, where A, B are convex sets of n^{1/2} real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n^{31/14}) unit-area triangles.
  • Combinatorial geometry
  • incidences
  • repeated configurations


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