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

^*Corresponding author for this work

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

4 Scopus citations

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
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2016 Elsevier B.V.

Keywords

Cobinatorial geometry
Convex sets
Repeated Configurations

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10.1016/j.comgeo.2016.12.002

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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

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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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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

Bibliographical note

Keywords

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