Heilbronn's triangle problem in three dimensions

Abstract

We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c>0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's triangle problem. This estimate is a consequence of the following result about configurations of point-line pairs in ${\mathbb{R}}^3$: for $n\ge 2$ let $p_1,\dots ,p_n\in [0,1{]}^3$ be a collection of points and let ${\ell }_i$ be a line through $p_i$ for every $i$ such that $d(p_i,{\ell }_j)\ge \hat{I}´$ for all $i\ne j$. Then we have $n\lesssim \hat{I}{´}^{-3+{\hat{I}}^3}$ for some absolute constant ${\hat{I}}^3>0$. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author.

34 pages