
How many points on the convex hull?
The convex hull of a set of points on the plane is, simply, the shape you'd get if you
stretched an elastic band around them. There is a more detailed explanation and
and an applet to help visualize the convex hull elsewhere on this site.
The number of points H on the convex hull depends on the number N and arrangement of the original points.
If we scatter points randomly on the plane, we can talk about the expected number of points
on the hull. This will depend on exactly how the original points are scattered.
The table below shows information about
the likely values of H for various values of N, for various distributions of points. Some explanation is in order.
 For each value of N, I randomly selected N points according to some distribution. Then I calculated the number of points H on the convex hull.
 In fact, I did this 100,000 times for each value of N.
 The table below shows
 the average value of H,
 the standard deviation
 the skewness and excess kurtosis (calculated from these formulae)
for each distribution of points.
 The four distributions of points are :
 Uniform square : the N points are uniformly distributed in a square (side length 2).
 Uniform circle : the N points are uniformly distributed in a circle (radius 1).
 Uniform triangle : the N points are uniformly distributed in an equilateral triangle.
 Gaussian : the N points are distributed on the plane according to a gaussian distribution.
 This is more useful than it appears at first. The results for the square apply to any
parallelogram, those for the circle to any ellipse, and those for the equilateral triangle
to any triangle at all. Similarly, for the gaussian distribution, my covariance matrix
is the identity, but the same results should apply for any bivariate gaussian distribution
of points.
n  Uniform Square  Uniform Circle  Uniform Eq. Triangle  Gaussian 
 mean  stdev  skew  kurt  mean  stdev  skew  kurt  mean  stdev  skew  kurt  mean  stdev  skew  kurt 
10  5.964  0.987  0.162  0.111  6.115  0.978  0.168  0.101  5.658  1.044  0.132  0.082  4.786  0.908  0.337  0.069 
20  7.754  1.305  0.183  0.037  8.266  1.267  0.208  0.017  7.092  1.348  0.181  0.0  5.476  1.053  0.351  0.114 
50  10.159  1.681  0.163  0.016  11.794  1.606  0.211  0.044  8.955  1.677  0.185  0.0020  6.327  1.207  0.359  0.136 
100  11.989  1.95  0.153  0.016  15.174  1.856  0.171  0.014  10.359  1.887  0.173  0.039  6.914  1.307  0.342  0.12 
200  13.829  2.182  0.147  0.017  19.356  2.126  0.173  0.035  11.747  2.083  0.159  0.02  7.465  1.396  0.339  0.087 
500  16.259  2.475  0.147  0.0010  26.547  2.528  0.138  0.0020  13.591  2.314  0.166  0.026  8.155  1.502  0.321  0.082 
1000  18.121  2.675  0.137  0.029  33.581  2.847  0.142  0.013  14.982  2.485  0.164  0.044  8.622  1.572  0.336  0.128 
2000  19.971  2.853  0.146  0.043  42.42  3.206  0.126  0.0030  16.345  2.624  0.15  0.021  9.089  1.631  0.317  0.069 
5000  22.4  3.092  0.131  0.056  57.716  3.752  0.094  0.0060  18.175  2.813  0.152  0.027  9.674  1.712  0.308  0.097 
10000  24.258  3.258  0.126  0.0020  72.794  4.219  0.088  0.0080  19.58  2.948  0.142  0.051  10.108  1.76  0.311  0.151 
As I mentioned, the above values are calculated from a sample of 100000 convex hulls for each value of N and distribution
of points. If anyone knows the theoretical "correct" values, please let me know!
Update! Thanks to Mike Beuoy for pointing out that I previously had the column hedings of the above table the wrong way round! This is now fixed!
Also, the theoretical correct values, at least for the means and variances, are known. For example, this Biometrika article (also available through JSTOR) gives integrals which give the mean and variance for the gaussian and uniform circle cases. Other articles give results for arbitrary polytopes  this would certainly cover the square and the triangle! Unfortunately, the integrals are very messy to work out.
