On the Asymptotic Distribution of the Area Outside a Random Convex Hull in a Disk

The asymptotic distribution of the area Vn outside the convex hull of n i.i.d. points uniformly distributed on the two-dimensional unit disk is studied. The asymptotic variance of Vn is found to be of the order n.5/3, and the asymptotic distribution of Vn is shown to be normal. The results are obtained by carefully analyzing the strength of dependence between sample points at different locations close to the boundary of the unit disk.