On the monotonicity of the moments of volumes of random simplices

TL;DR

This paper investigates the monotonicity properties of the moments of volumes of random simplices within convex bodies, revealing that these moments are generally not monotone under set inclusion, contrary to the expected behavior of the expected volume.

Contribution

It demonstrates that higher moments of the volume of random simplices do not follow monotonicity under set inclusion, extending previous work on the expected volume.

Findings

Expected volume is monotone under set inclusion.

Higher moments of volume are not monotone.

Counterexamples to monotonicity of moments.

Abstract

In a $d$-dimensional convex body $K$ random points $X_0, \dots, X_d$ are chosen. Their convex hull is a random simplex. The expected volume of a random simplex is monotone under set inclusion, if $K \subset L$ implies that the expected volume of a random simplex in $K$ is smaller than the expected volume of a random simplex in $L$. Continuing work of Rademacher, it is shown that moments of the volume of random simplices are in general not monotone under set inclusion.