Moments of volumes of lower-dimensional random simplices are not monotone

TL;DR

This paper investigates the monotonicity of moments of volumes of random lower-dimensional simplices within convex bodies, demonstrating that these moments are not monotone under set inclusion for certain dimensions, extending previous research.

Contribution

It extends prior work by showing that the moments of volumes of random simplices are not monotone under set inclusion when the simplex dimension is less than or equal to the ambient dimension.

Findings

Moments of volumes of random simplices are not monotone for n ≤ d.

Extends previous results by Rademacher and Reichenwallner-Reitzner.

Provides new insights into geometric probability and convex geometry.

Abstract

In a $d$-dimensional convex body $K$, for $n \leq d+1$, random points $X_0, \dots, X_{n-1}$ are chosen according to the uniform distribution in $K$. Their convex hull is a random $(n-1)$-simplex with probability $1$. We denote its $(n-1)$-dimensional volume by $V_{K[n]}$. The $k$-th moment of the $(n-1)$-dimensional volume of a random $(n-1)$-simplex is monotone under set inclusion, if $K \subseteq L$ implies that the $k$-th moment of $V_{K[n]}$ is not larger than that of $V_{L[n]}$. Extending work of Rademacher [On the monotonicity of the expected volume of a random simplex. Mathematika 58 (2012), 77--91] and Reichenwallner and Reitzner [On the monotonicity of the moments of volumes of random simplices. Mathematika 62 (2016), 949--958], it is shown that for $n \leq d$, the moments of $V_{K[n]}$ are not monotone under set inclusion.