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 $p_1, \ldots,p_n \in [0,1]^3$ be a collection of points and let $\ell_i$ be a line through $p_i$ for every $i$ such that $d(p_i, \ell_j) \ge δ$ for all $i\neq j$. Then we have $n \lesssim δ^{-3+γ}$ for some absolute constant $γ>0$. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author.