Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra

Abstract

Choose uniform random points in a given convex set and let be their convex hull. It is shown that in dimension three the expected volume of this convex hull is in general not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is formed by uniformly distributed points in the three-dimensional tetrahedron together with a small perturbation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.