The average‐case area of Heilbronn‐type triangles*

Abstract

AbstractFrom among \documentclass{article}\pagestyle{empty}\begin{document}${n \choose 3}$\end{document} 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/n3<μn<C/n3 for all large enough values of n, where μn is the expectation of A. Moreover, c/n3<A<C/n3, 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.” © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 206–219, 2002