Monotonicity of functionals of random polytopes

Abstract

The convex hull $P_n$ of a Gaussian sample $X_1,...,X_n$ in $R^d$ is a Gaussian polytope. We prove that the expected number of facets $E{f}_{d-1}(P_n)$ is monotonically increasing in $n$. Furthermore we prove this for random polytopes generated by uniformly distributed points in a $d$-dimensional ball.

References in corpus (2)

• Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points
• Monotonicity of the Sample Range of 3-D Data: Moments of Volumes of Random Tetrahedra