A new upper bound for the Heilbronn triangle problem
Combinatorics
2023-05-30 v1 Classical Analysis and ODEs
Metric Geometry
Abstract
For sufficiently large , we show that in every configuration of points chosen inside the unit square there exists a triangle of area less than . 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.
Cite
@article{arxiv.2305.18253,
title = {A new upper bound for the Heilbronn triangle problem},
author = {Alex Cohen and Cosmin Pohoata and Dmitrii Zakharov},
journal= {arXiv preprint arXiv:2305.18253},
year = {2023}
}