Monotonicity of the Sample Range of 3-D Data: Moments of Volumes of Random Tetrahedra

2016/12/06 by Kunis, Stefan, Reichenwallner, Benjamin, Reitzner, Matthias
#52A22 #60D05 #62H11 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1612.01893

Abstract

The sample range of uniform random points X₁, … , X_n chosen in a given convex set is the convex hull \rm conv[X₁, …, X_n]. It is shown that in dimension three the expected volume of the sample range is not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is the three-dimensional tetrahedron together with an infinitesimal variation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.

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