## The expected size of Heilbronn's triangles (1999)

Venue: | Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity |

Citations: | 4 - 3 self |

### BibTeX

@INPROCEEDINGS{Jiang99theexpected,

author = {Tao Jiang and Ming Li},

title = {The expected size of Heilbronn's triangles},

booktitle = {Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity},

year = {1999},

pages = {105--113}

}

### OpenURL

### 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 ∆ = O(1/n 2). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c log n/n 2 ≤ ∆ ≤ C/n 8/7−ǫ for every constant ǫ> 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 Θ(1/n 3); and (ii) the smallest triangle has area Θ(1/n 3) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity. 1

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