TY - JOUR

T1 - On the number of unit-area triangles spanned by convex grids in the plane

AU - Raz, Orit E.

AU - Sharir, Micha

AU - Shkredov, Ilya D.

N1 - Funding Information: Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation, and by the Israeli Centers for Research Excellence (I-CORE) program (Center No. 4/11). Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Publisher Copyright: © 2016 Elsevier B.V.

PY - 2017/4/1

Y1 - 2017/4/1

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(n37/17log2/17n) 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(n37/17log2/17n) 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

UR - http://www.scopus.com/inward/record.url?scp=85008414347&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.comgeo.2016.12.002

DO - https://doi.org/10.1016/j.comgeo.2016.12.002

M3 - Article

SN - 0925-7721

VL - 62

SP - 25

EP - 33

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

ER -