On the mean value of the area of a random polygon in a plane convex body

Abstract

Let K be a convex body in Euclidean space R d , d ≥2, with volume V ( K ) = 1, and n ≥ d +1 be a natural number. We select n independent random points y 1 , y 2 , …, y n from K (we assume they all have the uniform distribution in K ). Their convex hull co { y 1 , y 2 , …, y n } is a random polytope in K with at most n vertices. Consider the expected value of the volume of this polytope It is easy to see that if U : R d → R d is a volume preserving affine transformation, then for every convex body K with V ( K ) = 1, m ( K, n ) = m ( U ( K ), n ).