@inproceedings{afbb21d1318646bcadd48e71a3707f60,

title = "The Number of Unit-Area Triangles in the Plane: Theme and Variations",

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 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 ω(n2), for any triple of lines (it is always O(n2) in this case). (ii) We show that if S is a convex grid of the form A × B, where A, B are convex sets of n1/2 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(n31/14) unit-area triangles.",

keywords = "Combinatorial geometry, Incidences, Repeated configurations",

author = "Raz, {Orit E.} and Micha Sharir",

year = "2015",

month = jun,

day = "1",

doi = "10.4230/LIPIcs.SOCG.2015.569",

language = "American English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

pages = "569--583",

editor = "Janos Pach and Janos Pach and Lars Arge",

booktitle = "31st International Symposium on Computational Geometry, SoCG 2015",

address = "Germany",

note = "31st International Symposium on Computational Geometry, SoCG 2015 ; Conference date: 22-06-2015 Through 25-06-2015",

}