InforLorV4, Main, Merge, bibRecord, 001361

The monotonicity of $f$-vectors of random polytopes

Identifieur interne : 001361 ( Main/Merge ); précédent : 001360; suivant : 001362

The monotonicity of $f$-vectors of random polytopes

Auteurs : Olivier Devillers [France] ; Marc Glisse [France] ; Xavier Goaoc [France] ; Guillaume Moroz [France] ; Matthias Reitzner [Allemagne]

Source :

Electronic Communications in Probability [ 1083-589X ] ; 2013.

RBID : Hal:hal-00805690

Abstract

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E[f0(Kn)] is increasing in n. In dimension d≥3 we prove that if limn→∞ E[fd−1(Kn)]Anc=1 for some constants A and c>0 then the function n↦E[fd−1(Kn)] is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.

Url:

https://hal.inria.fr/hal-00805690

DOI: 10.1214/ECP.v18-2469

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<author><name sortKey="Reitzner, Matthias" sort="Reitzner, Matthias" uniqKey="Reitzner M" first="Matthias" last="Reitzner">Matthias Reitzner</name>
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<country>Allemagne</country>
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<idno type="DOI">10.1214/ECP.v18-2469</idno>
<series><title level="j">Electronic Communications in Probability</title>
<idno type="ISSN">1083-589X</idno>
<imprint><date type="datePub">2013</date>
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<front><div type="abstract" xml:lang="en">Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E[f0(Kn)] is increasing in n. In dimension d≥3 we prove that if limn→∞ E[fd−1(Kn)]Anc=1 for some constants A and c>0 then the function n↦E[fd−1(Kn)] is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.</div>
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The monotonicity of $f$-vectors of random polytopes

The monotonicity of $f$-vectors of random polytopes

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