The monotonicity of $f$-vectors of random polytopes

Olivier Devillers 1 Marc Glisse 1 Xavier Goaoc 2 Guillaume Moroz 2 Matthias Reitzner 3
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
3 Institut für Mathematik
FB6/Institut für Mathematik - Institut für Mathematik [Osnabrück]
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.
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Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, Matthias Reitzner. The monotonicity of $f$-vectors of random polytopes. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2013, 18 (23), pp.1-8. ⟨10.1214/ECP.v18-2469⟩. ⟨hal-00805690⟩

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