On the LLN for the number of vertices of a random convex hull

For several common parent laws, the number of vertices of a sample convex hull follows a kind of law of large numbers. We exhibit an example of a parent law which contradicts a general conjecture about this matter.

[1]  W. Reed,et al.  Random points in a simplex , 1974 .

[2]  Luc.  On the oscillation of the expected number of extreme points of a random set , 1990 .

[3]  Rolf Schneider,et al.  Random approximation of convex sets * , 1988 .

[4]  Piet Groeneboom,et al.  Limit theorems for convex hulls , 1988 .

[5]  Josef S. Müller.  Approximation of a ball by random polytopes , 1990 .

[6]  B. Efron.  The convex hull of a random set of points , 1965 .

[7]  Irene Hueter,et al.  The convex hull of a normal sample , 1994, Advances in Applied Probability.

[8]  Piet Groeneboom,et al.  Limit theorems for functionals of convex hulls , 1994 .

[9]  David Mannion,et al.  The Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Parent Tetrahedron , 1994, Advances in Applied Probability.

[10]  V. Alagar.  ON THE DISTRIBUTION OF A RANDOM TRIANGLE , 1977 .

[11]  Tailen Hsing.  On the Asymptotic Distribution of the Area Outside a Random Convex Hull in a Disk , 1994 .

[12]  Convergence in mean of some characteristics of the convex hull , 1989, Advances in Applied Probability.

[13]  J. F. C. Kingman,et al.  Random secants of a convex body , 1969, Journal of Applied Probability.

[14]  Christian Buchta,et al.  Distribution-independent properties of the convex hull of random points , 1990 .

[15]  David Aldous,et al.  The number of extreme points in the convex hull of a random sample , 1991 .

[16]  Random triangles in convex regions , 1983 .

[17]  Invariance principle for the deviation between the probability content and the interior point proportion of a random convex hull , 1995 .

[18]  Bruno Massé.  On the variance of the number of extreme points of a random convex hull , 1999 .