A power improvement in the Heilbronn triangle lower bound

For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least n^{-2+c} for one absolute c > 0. This disproves the conjectured almost-n⁻² upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be. (Lean)

Manuscript

A power improvement in the Heilbronn triangle lower bound

September 25, 2026 · source · cite as \cite{OAI:A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026}

There are absolute constants \eta,c_1\gt 0 such that, for every sufficiently large integer n, one can choose n points in the unit square so that every triangle they determine has area at least c_1n^{-2+\eta}. Thus the almost n⁻² upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.

Result 191 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.