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.
The paper gives a dimension-dependent condition for the monotonicity to hold (Lemma 12 in http://arxiv.org/abs/1008.3944v1).
In the paper, it is shown that the condition holds for dimensions 1 and
2, and does not hold for dimensions 4 and higher, while the 3
dimensional case is left open. In the 3-dimensional case, a simplex and
center of one of its facets can be verified numerically (Mathematica source) to be a counterexample to the dimension-dependent condition, suggesting that the monotonicity does not hold in 3 dimensions.