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2013 The monotonicity of $f$-vectors of random polytopes
Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, Matthias Reitzner
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Electron. Commun. Probab. 18: 1-8 (2013). DOI: 10.1214/ECP.v18-2469

Abstract

Let $K$ be a compact convex body in ${\mathbb R}^d$, let $K_n$ be the convex hull of $n$ points chosen uniformly and independently in $K$, and let $f_{i}(K_n)$ denote the number of $i$-dimensional faces of $K_n$. ;We show that for planar convex sets, $E[f_0 (K_n)]$ is increasing in $n$. In dimension $d \geq 3$ we prove that if $\lim_{n \to \infty} \frac{E[f_{d-1}(K_n)]}{An^c}=1$ for some constants $A$ and $c>0$ then the function $n \mapsto E[f_{d-1}(K_n)]$ 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." Electron. Commun. Probab. 18 1 - 8, 2013. https://doi.org/10.1214/ECP.v18-2469

Information

Accepted: 28 March 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 1359.60024
MathSciNet: MR3044471
Digital Object Identifier: 10.1214/ECP.v18-2469

Subjects:
Primary: 60D05
Secondary: 52A22

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