A power improvement in the Heilbronn triangle lower bound

What OpenAI claims

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)

Paper

1. A power improvement in the Heilbronn triangle lower bound

   25 September 2026 · Preprint, not peer reviewed · PDF

   There are absolute constants η,c₁ > 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 c₁n^-2+η. Thus the almost n−2 upper-bound formulation of Heilbronn's triangle problem is false. The exponent η is fixed but extremely small.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.