License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.569
URN: urn:nbn:de:0030-drops-51125
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5112/
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Raz, Orit E. ; Sharir, Micha

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

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Abstract

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.

BibTeX - Entry

@InProceedings{raz_et_al:LIPIcs:2015:5112,
  author =	{Orit E. Raz and Micha Sharir},
  title =	{{The Number of Unit-Area Triangles in the Plane: Theme and Variations}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{569--583},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5112},
  URN =		{urn:nbn:de:0030-drops-51125},
  doi =		{10.4230/LIPIcs.SOCG.2015.569},
  annote =	{Keywords: Combinatorial geometry, incidences, repeated configurations}
}

Keywords: Combinatorial geometry, incidences, repeated configurations
Collection: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 12.06.2015

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