A power improvement in the Heilbronn triangle lower bound
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
A power improvement in the Heilbronn triangle lower bound
There are absolute constants \eta,c_1> 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…
Full source abstract
There are absolute constants \eta,c_1> 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.
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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.