The average-case area of Heilbronn-type triangles
Random Structures & Algorithms
SOFSEM '00 Proceedings of the 27th Conference on Current Trends in Theory and Practice of Informatics
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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.