New Lower Bounds for Heilbronn Numbers

Abstract

The $n$-th Heilbronn number, $H_n$, is the largest value such that $n$ points can be placed in the unit square in such a way that all possible triangles defined by any three of the points have area at least $H_n$. In this note we establish new bounds for the first Heilbronn numbers. These new values have been found by using a simple implementation of simulated annealing to obtain a first approximation and then optimizing the results by finding the nearest exact local maximum.