Heilbronn's triangle problem in three dimensions

We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2\mathrm{/}3-c}$ 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 ${\mathrm{\ell }}_i$ be a line through $p_i$ for every $i$ such that $d(p_i,{\mathrm{\ell }}_j)\ge \delta$ for all $i\mathrm{\ne }j$. Then we have $n\lesssim {\delta }^{-3+\gamma }$ for some absolute constant $\gamma >0$. The analogous result about point-line configurations in the plane was previously established by Cohen, Pohoata and the last author.