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The Average-Case Area of Heilbronn-Type Triangles

Tao Jiang·Ming Li·Paul Vitanyi
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Abstract

From among $ {n \choose 3}$ triangles with vertices chosen from $n$ points in the unit square, let $T$ be the one with the smallest area, and let $A$ be the area of $T$. Heilbronn's triangle problem asks for the maximum value assumed by $A$ over all choices of $n$ points. We consider the average-case: If the $n$ points are chosen independently and at random (with a uniform distribution), then there exist positive constants $c$ and $C$ such that $c/n^3 < μ_n < C/n^3$ for all large enough values of $n$, where $μ_n$ is the expectation of $A$. Moreover, $c/n^3 < A < C/n^3$, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in ``general position.''

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Tao Jiang, Ming Li, Paul Vitanyi. 2003-11-10. The Average-Case Area of Heilbronn-Type Triangles. https://arxiv.org/abs/math/9902043

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