A new upper bound for the Heilbronn triangle problem

2023/05/29 by Alex S. Cohen, Cosmin Pohoata, Cohen, Alex +3 · 3 citations
Mathematics · Computer Science · #Point processes and geometric inequalities #Computational Geometry and Mesh Generation #Advanced Graph Theory Research

paper · pdf · doi:10.48550/arxiv.2305.18253

Abstract

For sufficiently large n, we show that in every configuration of n points chosen inside the unit square there exists a triangle of area less than n^-8/7-1/2000. This improves upon a result of Komlós, Pintz and Szemerédi from 1982. Our approach establishes new connections between the Heilbronn triangle problem and various themes in incidence geometry and projection theory which are closely related to the discretized sum-product phenomenon.

Cited by

• Heilbronn's triangle problem in three dimensions
• Point-Line Incidence Estimates in (ℤ/p^kℤ)²
• On subsets of lattice cubes avoiding affine and spherical degeneracies

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