Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317) (1999)
May 4, 1999 to May 6, 1999
Tao Jiang , McMaster University
Ming Liy , University of Waterloo
Paul Vitányi , CWI and University of Amsterdam
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.
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.