Central limit theorem for the volume of random polytopes with vertices on the boundary
type: Item
instance of: scholarly article
title: Central limit theorem for the volume of random polytopes with vertices on the boundary
zbMATH Open document ID: 1400.52005
author: Christoph Thäle
published in: Discrete \& Computational Geometry
publication date: 2018-07-13T00:00:00Z
review text: Denote by \({\mathcal{K}}_2^+\) the space of convex bodies \(K\subset\mathbb{R}^d\), \(d\geq 2\) with volume \(\mathrm{vol}_d(K)=1\), which have a boundary \(\partial K\) that is twice differentiable and has strictly positive Gaussian curvature at every point. Let \({{\mathcal H}}^{d-1}\) be the \((d-1)\)-dimensional normalized Hausdorff measure on \(\partial K\) (\({{\mathcal H}}^{d-1} (\partial K) = 1\)). For a fixed integer \(n\geq d+1\) let \(X_1, X_2,\dots, X_n\) be independent random points that are chosen from \(\partial K\) with respect to the probability distribution \({\mathcal{H}}^{d-1}\). The random convex hull \(K_n =\mathrm{conv}\{X_1, X_2,\dots, X_n\}\) is a random polytope in \(K\) having all its vertices on the boundary \(\partial K\). \textit{R. M. Richardson} et al. [Eur. J. Comb. 28, No. 8, 2057--2071 (2007; Zbl 1165.60008); Discrete Comput. Geom. 39, No. 1--3, 469--499 (2008; Zbl 1148.52006)] provided a central limit theorem, but only if a further randomization is added to the model. In [Zbl 1165.60008] and [Zbl 1148.52006] a Poisson random polytopes on the boundary of \(K\) is considered, where the fixed number of vertices \(n\) is replaced by a random number of vertices that follows a Poisson distribution with mean \(n\). The effect of this additional randomization is a further independence property. Namely, if it is known that a certain region in the boundary of \(K\) contains \(k\leq n\) points, say, then the remaining part has to contain exactly \(n-k\) points in the model. On the other hand, the number of vertices in that region and its complement are independent random variables in the Poissonized model. This construction finally allows to build on Stein's method for weakly dependent random variables, which leads to a central limit theorem for the volume of the Poisson random polytopes on the boundary of \(K\). For classical random polytopes, where the points are selected from the interior of the convex body \(K\) according to the uniform…
reviewed by: Viktor Ohanyan
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 6903508
zbMATH Keywords: central limit theorem
zbMATH Keywords: random polytope
zbMATH Keywords: surface body
zbMATH Keywords: stochastic geometry
MaRDI profile type: Publication
full work available at URL: s00454-017-9862-2
OpenAlex ID: W2587174486
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DOI: 10.1007/S00454-017-9862-2
description: scientific article; zbMATH DE number 6903508