The expected size of Heilbronn's triangles

Abstract

Heilbronn's triangle problem asks for the least such that n points lying in the unit disc necessarily contain a triangle of area at most . Heilbronn initially conjectured = O1=n 2 . As a result of concerted mathematical e ort it is currently known that there are positive constants c and C such that c log n=n 2 C=n 8=7, for every constant 0.We resolve Heilbronn's problem in the expected case:If we uniformly at random put n points in the unit disc then i the area of the smallest triangle has expectation 1=n 3 ; and ii the smallest triangle has area 1=n 3 with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity.

New Lower Bounds for Heilbronn Numbers

“…Recent approaches to the Heilbronn problem include an algorithm, provided in 1997 by C. Bertram-Kretzberg, T. Hofmeister, and H. Lefmann [3], which for a discretization of the problem finds a triangle with area log n/n 2 for every fixed n; lower bounds on higher dimensional versions of the problem, produced in 1999 by G. Barequet [2] and the study of the average size of the triangles by T. Jiang, M. Li, and P. Vitányi [8,9].…”

The average‐case area of Heilbronn‐type triangles*

“…John Tromp has informed us in December 1999 that, following a preliminary version[10] of this work, he has given an alternative proof of the main result based on the probabilistic method.…”