Central limit theorem for the volume of random polytopes with vertices on the boundary
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 distribution, it is well known from [\textit{M. Reitzner}, Probab. Theory Relat. Fields 133, No. 4, 483--507 (2005; Zbl 1081.60008)] and [\textit{V. Vu}, Adv. Math. 207, No. 1, 221--243 (2006; Zbl 1111.52010)] that a coupling can be constructed to push the central limit theorem for the Poissonized model to the original one. However, no such coupling is possible for the random polytopes \(K_n\) (see [Zbl 1165.60008] and [Zbl 1148.52006]). The reason of this phenomenon is that a Poisson random variable with mean \(n\) has fluctuations typically of order \(\sqrt{n}\), but the contribution of \(\sqrt{n}\) points on \(\partial K\) to the volume of the random polytope are of the same order as the variance. Therefore, a central limit theorem for the volume of \(K_n\) has been left as a major open problem in [Zbl 1165.60008] and [Zbl 1148.52006]. The main goal of this paper is to solve this problem and to prove the following central limit theorem with a bound on the rate convergence. Theorem. Let \(K\in {\mathcal{K}}_2^+\) and \(N\) be the standard Gaussian random variable. Then there exist constants \(n_0\) (natural) and \(c\in(0, \infty)\) only depending on \(d\) and on \(K\) such that \[ \sup_{u\in\mathbb{R}} \Big|P\Big({{\mathrm{vol}_d(K_n) - \mathrm{Evol}_d(K_n)}\over {\sqrt{\mathrm{Var} (\mathrm{vol}_d(K_n))}}} \leq u\Big) - P(N\leq u)\Big| \leq c n^{-1/2} (\ln n)^{3{{d+1}\over{d-1}}}\text{ for all }n\geq n_0. \] In particular, as \(n\to \infty\), one has that \({{\mathrm{vol}_d(K_n) - \mathrm{Evol}_d(K_n)} \over {\sqrt{\mathrm{Var} (\mathrm{vol}_d(K_n))}}}\) tends to \(N\) in distribution (\(E[\cdot]\) is the expectation and \(\mathrm{Var}[\cdot]\) is the variance).
scientific article; zbMATH DE number 6903508
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