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Random polytopes and the wet part for arbitrary probability distributions

Abstract : We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for the uniform distribution from a convex set [B\'ar\'any and Larman 1988]. The lower bound of B\'ar\'any and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of $\log n$. We show by an example that this is tight.
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Contributor : Xavier Goaoc Connect in order to contact the contributor
Submitted on : Monday, September 14, 2020 - 10:11:25 AM
Last modification on : Friday, February 4, 2022 - 9:00:13 AM

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Imre Barany, Matthieu Fradelizi, Xavier Goaoc, Alfredo Hubard, Günter Rote. Random polytopes and the wet part for arbitrary probability distributions. Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2020, 3, pp.701-715. ⟨10.5802/ahl.44⟩. ⟨hal-02937527⟩



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