Moments of volumes of lower-dimensional random simplices are not monotone

摘要

In a d-dimensional convex body K, for , random points are chosen according to the uniform distribution in K. Their convex hull is a random -simplex with probability 1. We denote its -dimensional volume by . The k-th moment of the -dimensional volume of a random -simplex is monotone under set inclusion, if implies that the k-th moment of is not larger than that of . Extending work of Rademacher (Mathematika 58:77-91, 2012) and Reichenwallner and Reitzner (Mathematika 62:949-958, 2016), it is shown that for , the moments of are not monotone under set inclusion. As a consequence, the nonmonotonicity of the expected surface area of the convex hull of uniform random points in a d-dimensional convex body follows.