Exact formulae for variances of functionals of convex hulls

Abstract

The vertices of the convex hull of a uniform sample from the interior of a convex polygon are known to be concentrated close to the vertices of the polygon. Furthermore, the remaining area of the polygon outside of the convex hull is concentrated close to the vertices of the polygon. In order to see what happens in a corner of the polygon given by two adjacent edges, we consider - in view of affine invariance - n points P₁,..., P_n distributed independently and uniformly in the interior of the triangle with vertices (0, 1), (0, 0), and (1, 0). The number of vertices of the convex hull, which are close to the origin (0, 0), is then given by the number Ñ_n of points among P₁,..., P_n, which are vertices of the convex hull of (0, 1), P₁,..., P_n, and (1, 0). Correspondingly, D̃_n is defined as the remaining area of the triangle outside of this convex hull. We derive exact (nonasymptotic) formulae for varÑ_n and varD̃_n. These formulae are in line with asymptotic distribution results in Groeneboom (1988), Nagaev and Khamdamov (1991), and Groeneboom (2012), as well as with recent results in Pardon (2011), (2012).