Heilbronn's triangle problem in three dimensions

TL;DR

This work extends Heilbronn's triangle problem to three dimensions by proving Δ_{3,3}(n) ≤ C n^{-2/3 - c} for some absolute c>0, establishing the first nontrivial 3D upper bound. The authors reduce the problem to a 3D point-line incidence framework, developing a robust high-low methodology that leverages two-ends and hairbrush techniques, and they connect the 3D bound to a planar PL_2(γ) result to obtain PL_3(κ) with κ>0. Building on Cohen–Pohoata and recent harmonic-analysis methods, they craft a comprehensive incidence-geometry toolkit—incidence decompositions, Fourier-analytic two-ends, rescaling arguments, and Katz–Tao-like concentration bounds—that translates 2D control into a 3D bound. Although the explicit c is tiny and not optimized, the paper introduces novel analytic machinery with potential for further refinements and generalizations to higher 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.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2: cohen2025lower
• Theorem 1.3
• proof : Proof of Theorem \ref{['thm:triangles-in-R3']}
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Lemma 3.1
• proof