To illustrate some examples of the "weirdness" of geometric probability, consider the following three results:

- If we pick n points at random from the unit square, the expected size of the convex hull of these points is log n.
- If we pick n points at random from a k-gon, the expected size of the convex hull is k log n
- If we pick n points at random from a unit disk, the expected size of the convex hull is n
^{1/3}!

But a result that I found even more surprising is this one, by Golin, Langerman and Steiger.

An arrangement of n lines in RTheir result uses a fact that is well known (Feller vol. 2), but is still remarkably elegant.^{2}induces a vertex set (all intersection points) whose convex hull has expected complexity O(1) !

Choose points xSpecifically this means that the minimum is distributed as an exponential distribution, and the rest are distributed as the appropriate gamma distributions._{1}, x_{2}, ..., x_{n}at random on the unit interval. Compute their sorted order x_{(1)}, x_{(2)}, ... x_{(n)}, the order statistic. Then the variable x(k) is distributed (roughly) as the sum of k i.i.d exponentially distributed random variables.

Bill Steiger recently gave a talk at the NYU Geometry seminar. I could not attend, but the abstract of his talk indicates that the above result holds even in R

^{3}.

Update: I should clarify that the O(1) result mentioned above was first proved by Devroye and Toussaint; their distribution on lines is different, and their proof is more complex. The above paper presents a simpler proof for an alternate distribution.