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arXiv · 1008.3944

On the monotonicity of the expected volume of a random simplex

Luis Rademacher
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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.

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Luis Rademacher. 2010-08-24. On the monotonicity of the expected volume of a random simplex. https://doi.org/10.1112/s0025579311002063

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