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

2015/04/27 by Orit E. Raz, Micha Sharir, Raz, Orit E. +3
Computer Science · Mathematics · #11B30 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Point processes and geometric inequalities #math.CO #msc:11B30

paper · pdf · doi:10.48550/arxiv.1504.06989

arXiv admin note: substantial text overlap with arXiv:1501.00379

arxiv created 2015/04/27 · openalex publication_date 2015/04/27 · arxiv updated 2015/04/28 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

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⊂\mathbb R, each of size n^1/2, the convex grid A× B spans at most O(n^37/17log^2/17n) unit-area triangles. This improves the best known upper bound O(n^31/14) recently obtained in \citeRS. Our analysis also applies to more general families of sets A, B, known as sets of Szemerédi--Trotter type.

Citations

• The number of unit-area triangles in the plane: Theme and variations

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