31 research outputs found
The Average-Case Area of Heilbronn-Type Triangles
From among triangles with vertices chosen from points in
the unit square, let be the one with the smallest area, and let be the
area of . Heilbronn's triangle problem asks for the maximum value assumed by
over all choices of points. We consider the average-case: If the
points are chosen independently and at random (with a uniform distribution),
then there exist positive constants and such that for all large enough values of , where is the expectation of
. Moreover, , with probability close to one. Our proof
uses the incompressibility method based on Kolmogorov complexity; it actually
determines the area of the smallest triangle for an arrangement in ``general
position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a
roll, {\em New Scientist}, November 6, 1999, 44--4