Monotonicity of functionals of random polytopes

TL;DR

This paper proves that the expected number of facets of Gaussian and uniform random polytopes increases monotonically with the number of points, providing new insights into their geometric properties.

Contribution

It establishes the monotonicity of the expected number of facets for Gaussian and uniform ball-generated random polytopes, a novel result in stochastic geometry.

Findings

Expected number of facets increases with sample size for Gaussian polytopes.

Expected number of facets increases with sample size for uniform ball polytopes.

Monotonicity holds in both Gaussian and uniform ball settings.

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.