标题： 随机边界多面体的中心极限定理 显示英文标题

标题： Central limit theorems for random boundary polytopes

摘要： 凸包的面数，其中$n$个独立同分布的随机点是在$\mathbb{R}^d$维光滑凸体的边界上选择的。在二维和三维中，$k$-面的数量几乎可以确定是常数，在四维及更高维中，如果$k\ge 1$，方差已知是非零的。我们证明它是$n$的数量级。这由一个中心极限定理补充，其 Berry-Esseen 界是最佳阶数$n^{-1/2}$。我们导出了类似的关于泊松化模型的结果，其中额外的随机点数量是泊松分布的。作为主要工具，我们开发了面数作为一个指数稳定得分函数之和的表示。 显示英文摘要

摘要： The number of faces of the convex hull of $n$ independent and identically distributed random points chosen on the boundary of a smooth convex body in $\mathbb{R}^d$ is investigated. In dimensions two and three the number of $k$-faces is known to be constant almost surely and in dimension four and higher the variance is known to be non-zero if $k\ge 1$. We show that it is of order $n$. This is complemented by a central limit theorem with a Berry-Esseen bound which is of optimal order $n^{-1/2}$. We derive similar results for the Poissonized model, where additionally the number of random points is Poisson distributed. As a main tool, we develop a representation of the number of faces as a sum of exponentially stabilizing score functions.