The Average-Case Area of Heilbronn-Type Triangles
Abstract
From among
(
n
3
)
triangles with vertices chosen from
n
points in the unit square, let
T
be the one with the smallest area, and let
A
be the area of
T
. Heilbronn's triangle problem asks for the maximum value assumed by
A
over all choices of
n
points. We consider the average-case: If the
n
points are chosen independently and at random (with a uniform distribution), then there exist positive constants
c
and
C
such that
c/
n
3
<
μ
n
<C/
n
3
for all large enough values of
n
, where
μ
n
is the expectation of
A
. Moreover,
c/
n
3
<A<C/
n
3
, 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.''