On the variance of random polygons

Abstract

A random polygon is the convex hull of uniformly distributed random points in a convex body K⊂R2. General upper bounds are established for the variance of the area of a random polygon and also for the variance of its number of vertices. The upper bounds have the same order of magnitude as the known lower bounds on variance for these functionals. The results imply a strong law of large numbers for the area and number of vertices of random polygons for all planar convex bodies. Similar results had been known, but only in the special cases when K is a polygon or where K is a smooth convex body. The careful, technical arguments we needed may lead to tools for analogous extensions to general convex bodies in higher dimension. On the other hand one of the main results is a stronger version in dimension d=2 of the economic cap covering theorem of Bárány and Larman. It is crucial to our proof, but it does not extend to higher dimension.