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Journal Articles Annales Henri Lebesgue Year : 2020

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.

Dates and versions

hal-02937527 , version 1 (14-09-2020)

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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, 2020, 3, pp.701-715. ⟨10.5802/ahl.44⟩. ⟨hal-02937527⟩
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