  Using ideas from the geometry of compression, we improve on the current upper
and lower bound of Heilbronn's triangle problem. In particular, by letting
$\Delta(s)$ denotes the minimal area of the triangle induced by $s$ points in a
unit disc, then 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$$\Delta(s)\gg \frac{\log s}{s\sqrt{s}}.$$
