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arXiv · 2006.05269

New bounds for the Heilbronn triangle problem

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

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Theophilus Agama. 2020-06-05. New bounds for the Heilbronn triangle problem. https://arxiv.org/abs/2006.05269

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