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New lower bounds for Heilbronn numbers. (English) Zbl 1008.52020

Electron. J. Comb. 9, No. 1, Research paper R6, 10 p. (2002); printed version J. Comb. 9, No. 1 (2002).
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 \(n\) points have an area of at least \(H\). In this paper the authors find new lower bounds for \(H_7\), \(H_8\), \(H_9\), \(H_{10}\) and \(H_{12}\). In order to derive these bounds a simulated annealing algorithm is used resulting in a good configuration of the points. This solution is then refined by using an analytical procedure to get the closest local maximum.

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
51M25 Length, area and volume in real or complex geometry
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