Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference
DOI: 10.1109/ccc.1999.766269
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The expected size of Heilbronn's triangles

Abstract: Heilbronn's triangle problem asks for the least such that n points lying in the unit disc necessarily contain a triangle of area at most . Heilbronn initially conjectured = O1=n 2 . As a result of concerted mathematical e ort it is currently known that there are positive constants c and C such that c log n=n 2 C=n 8=7, for every constant 0.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 1=n 3 ; … Show more

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Cited by 3 publications
(2 citation statements)
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“…Recent approaches to the Heilbronn problem include an algorithm, provided in 1997 by C. Bertram-Kretzberg, T. Hofmeister, and H. Lefmann [3], which for a discretization of the problem finds a triangle with area log n/n 2 for every fixed n; lower bounds on higher dimensional versions of the problem, produced in 1999 by G. Barequet [2] and the study of the average size of the triangles by T. Jiang, M. Li, and P. Vitányi [8,9].…”
Section: Introductionmentioning
confidence: 99%
“…Recent approaches to the Heilbronn problem include an algorithm, provided in 1997 by C. Bertram-Kretzberg, T. Hofmeister, and H. Lefmann [3], which for a discretization of the problem finds a triangle with area log n/n 2 for every fixed n; lower bounds on higher dimensional versions of the problem, produced in 1999 by G. Barequet [2] and the study of the average size of the triangles by T. Jiang, M. Li, and P. Vitányi [8,9].…”
Section: Introductionmentioning
confidence: 99%
“…John Tromp has informed us in December 1999 that, following a preliminary version[10] of this work, he has given an alternative proof of the main result based on the probabilistic method.…”
mentioning
confidence: 99%