On the number of unit-area triangles spanned by convex grids in the plane

Orit E. Raz; Micha Sharir; Ilya D. Shkredov

doi:10.1016/j.comgeo.2016.12.002

On the number of unit-area triangles spanned by convex grids in the plane

Orit E. Raz, Micha Sharir, Ilya D. Shkredov

School of Computer Science

Tel Aviv University

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

Abstract

A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets A,B⊂R, each of size n^1/2, the convex grid A×B spans at most O(n^37/17log^2/17⁡n) unit-area triangles. Our analysis also applies to more general families of sets A, B, known as sets of Szemerédi–Trotter type.

Original language	English
Pages (from-to)	25-33
Number of pages	9
Journal	Computational Geometry: Theory and Applications
Volume	62
DOIs	https://doi.org/10.1016/j.comgeo.2016.12.002
State	Published - 1 Apr 2017

Keywords

Cobinatorial geometry
Convex sets
Repeated Configurations

All Science Journal Classification (ASJC) codes

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.comgeo.2016.12.002

Cite this

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abstract = "A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets A,B⊂R, each of size n1/2, the convex grid A×B spans at most O(n37/17log2/17⁡n) unit-area triangles. Our analysis also applies to more general families of sets A, B, known as sets of Szemer{\'e}di–Trotter type.",

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AU - Sharir, Micha

AU - Shkredov, Ilya D.

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N2 - A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets A,B⊂R, each of size n1/2, the convex grid A×B spans at most O(n37/17log2/17⁡n) unit-area triangles. Our analysis also applies to more general families of sets A, B, known as sets of Szemerédi–Trotter type.

AB - A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets A,B⊂R, each of size n1/2, the convex grid A×B spans at most O(n37/17log2/17⁡n) unit-area triangles. Our analysis also applies to more general families of sets A, B, known as sets of Szemerédi–Trotter type.

KW - Cobinatorial geometry

KW - Convex sets

KW - Repeated Configurations

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DO - 10.1016/j.comgeo.2016.12.002

M3 - مقالة

SN - 0925-7721

VL - 62

SP - 25

EP - 33

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

ER -

On the number of unit-area triangles spanned by convex grids in the plane

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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