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 (4)

• Brownian limits, local limits and variance asymptotics for convex hulls in the ball
• 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
• Monotonicity of facet numbers of random convex hulls

Cited by in corpus (1)

• Random polytopes and the wet part for arbitrary probability distributions