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Reports (Research Report) Year : 2019

Random polytopes and the wet part for arbitrary probability distributions

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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.

Dates and versions

hal-02050632 , version 1 (27-02-2019)

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Imre Barany, Matthieu Fradelizi, Xavier Goaoc, Alfredo Hubard, Günter Rote. Random polytopes and the wet part for arbitrary probability distributions. [Research Report] Rényi Institute of Mathematics; University College London; Université Paris-Est; Université de Lorraine; Freie Universität Berlin. 2019. ⟨hal-02050632⟩
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