On the monotonicity of the expected volume of a random simplex

TL;DR

This paper investigates whether the expected volume of a random simplex within a convex body increases with the body’s size, confirming monotonicity in low dimensions and disproving it in higher dimensions, with implications for the slicing conjecture.

Contribution

The paper establishes monotonicity results for the expected volume of random simplices in dimensions 1 and 2, and shows non-monotonicity in dimensions 4 and above, also analyzing higher moments.

Findings

Monotonicity holds for dimensions 1 and 2.

Monotonicity does not hold for dimensions 4 and higher.

Results relate to the slicing conjecture and higher moments of volume.

Abstract

Let a random simplex in a d-dimensional convex body be the convex hull of d+1 random points from the body. We study the following question: As a function of the convex body, is the expected volume of a random simplex monotone non-decreasing under inclusion? We show that this holds if d is 1 or 2, and does not hold if d >= 4. We also prove similar results for higher moments of the volume of a random simplex, in particular for the second moment, which corresponds to the determinant of the covariance matrix of the convex body. These questions are motivated by the slicing conjecture.