The convex hull of a normal sample

Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices N n , the perimeter L n and the area A n of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of N n , L n and A n for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.