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تسجيل مستخدم جديدMonotonicity of the Sample Range of 3-D Data: Moments of Volumes of Random Tetrahedra
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The sample range of uniform random points $X_1, dots , X_n$ chosen in a given convex set is the convex hull ${rm conv}[X_1, dots, X_n]$. 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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