Electronic Communications in Probability

The monotonicity of $f$-vectors of random polytopes

Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, and Matthias Reitzner

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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.

Article information

Electron. Commun. Probab. Volume 18 (2013), paper no. 23, 8 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]

Computational geometry Convex hull Complexity

This work is licensed under a Creative Commons Attribution 3.0 License.


Devillers, Olivier; Glisse, Marc; Goaoc, Xavier; Moroz, Guillaume; Reitzner, Matthias. The monotonicity of $f$-vectors of random polytopes. Electron. Commun. Probab. 18 (2013), paper no. 23, 8 pp. doi:10.1214/ECP.v18-2469. https://projecteuclid.org/euclid.ecp/1465315562

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