New bounds for the Heilbronn triangle problem

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 $$ \Delta(s)\gg \frac{\log s}{s\sqrt{s}}. $$