The Expected Size of Heilbronn's Triangles
نویسندگان
چکیده
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). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c logn/n ≤ ∆ ≤ C/n 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); and (ii) the smallest triangle has area Θ(1/n) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity.
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