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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.

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Sep 25, 2026·main.pdf

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.

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