A new upper bound for the Heilbronn triangle problem

TL;DR

This paper presents an improved upper bound for the Heilbronn triangle problem, demonstrating that for large n, any configuration of points in the unit square contains a very small-area triangle, connecting geometric and combinatorial theories.

Contribution

The paper introduces a novel upper bound for the Heilbronn triangle problem and links it to incidence geometry and sum-product phenomena, advancing theoretical understanding.

Findings

Established a new upper bound of n^{-8/7-1/2000} for triangle areas

Connected Heilbronn problem to incidence geometry and sum-product theory

Improved upon the 1982 result by Komlós, Pintz, and Szemerédi

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 Koml\'os, Pintz and Szemer\'edi 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.