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

Orit E. Raz; Micha Sharir

doi:10.1007/s00493-016-3440-8

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

Orit E. Raz^*, Micha Sharir

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

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 [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).

Original language	English
Pages (from-to)	1221-1240
Number of pages	20
Journal	Combinatorica
Volume	37
Issue number	6
DOIs	https://doi.org/10.1007/s00493-016-3440-8
State	Published - 1 Dec 2017
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2016, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

Access to Document

10.1007/s00493-016-3440-8

Cite this

@article{665038cb001f472497ec95974fac75ff,

title = "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).",

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

note = "Publisher Copyright: {\textcopyright} 2016, J{\'a}nos Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.",

year = "2017",

month = dec,

day = "1",

doi = "10.1007/s00493-016-3440-8",

language = "אנגלית",

volume = "37",

pages = "1221--1240",

journal = "Combinatorica",

issn = "0209-9683",

publisher = "Janos Bolyai Mathematical Society",

number = "6",

}

TY - JOUR

T1 - The Number of Unit-Area Triangles in the Plane

T2 - Theme and Variation

AU - Raz, Orit E.

AU - Sharir, Micha

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84997831601&partnerID=8YFLogxK

U2 - 10.1007/s00493-016-3440-8

DO - 10.1007/s00493-016-3440-8

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84997831601

SN - 0209-9683

VL - 37

SP - 1221

EP - 1240

JO - Combinatorica

JF - Combinatorica

IS - 6

ER -

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

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this