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Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317) (1999)
Atlanta, Georgia
May 4, 1999 to May 6, 1999
ISSN: 1093-0159
ISBN: 0-7695-0075-7
pp: 105
Tao Jiang , McMaster University
Ming Liy , University of Waterloo
Paul Vitányi , CWI and University of Amsterdam
ABSTRACT
Heilbronn's triangle problem asks for the least \mathsuch that n points lying in the unit disc necessarily contain a triangle of area at most \math. Heilbronn initially conjectured \math. As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that \mathfor every constant \math.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 \math(1=n3 ); and (ii) the smallest triangle has area \math(1=n3 ) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity.
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CITATION

P. Vitányi, M. Liy and T. Jiang, "The Expected Size of Heilbronn's Triangles," Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)(CCC), Atlanta, Georgia, 1999, pp. 105.
doi:10.1109/CCC.1999.766269
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