New bounds for the Heilbronn triangle problem

TL;DR

This paper advances the bounds on the minimal triangle area formed by points on a unit disk, using geometric compression techniques to refine existing estimates.

Contribution

It introduces improved upper and lower bounds for the Heilbronn triangle problem leveraging geometric compression ideas.

Findings

Upper bound: Δ(s) ≪ 1 / s^{3/2 - ε} for small ε

Lower bound: Δ(s) ≫ (log s) / (s√s)

Enhanced understanding of minimal triangle areas in geometric configurations

Abstract

Using ideas from the geometry of compression, we improve on the current upper and lower bounds of the Heilbronn triangle problem. In particular, let $\Delta(s)$ denote the minimal area of the triangle induced by $s$ points on a unit disk. We have the upper bound $$ \Delta(s)\ll \frac{1}{s^{\frac{3}{2}-\epsilon}} $$ for small $\epsilon:=\epsilon(s)>0$ and the lower bound $$ \Delta(s)\gg \frac{\log s}{s\sqrt{s}}. $$