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arXiv · 1211.7020

The monotonicity of f-vectors of random polytopes

Olivier Devillers·Marc Glisse·Xavier Goaoc·Guillaume Moroz·Matthias Reitzner
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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 lim(E((f[d -1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the function E(f[d-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. 2012-11-29. The monotonicity of f-vectors of random polytopes. https://arxiv.org/abs/1211.7020

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