arXiv 1707.022532017-07math.MGmath.PR

随机β-多面体的期望内禀体积和面数

Expected intrinsic volumes and facet numbers of random beta-polytopes

Zakhar Kabluchko, Daniel Temesvari, Christoph Thaele

摘要：

设$X_1,\ldots,X_n$为i.i.d.\根据以下概率密度之一采样的$d$维欧几里得空间中的随机点：$$ f_{d,β} (x) = \text{const} \cdot (1-\|x\|^2)^β, \quad \|x\|\leq 1, \quad \text{(the beta case)} $$和$$ \tilde f_{d,β} (x) = \text{const} \cdot (1+\|x\|^2)^{-β}, \quad x\in\mathbb{R}^d, \quad \text{(the beta' case).} $$我们精确地计算$X_1,\ldots,X_n$的凸包的期望的固有体积和期望的刻面数。渐近公式，以前获得Affentanger [随机点的凸包与球对称分布，1991年]。通过研究当$β\downarrow -1$，分别为$β\uparrow +\infty$时β情况的极限，我们还可以分别覆盖其中$X_1,\ldots,X_n$在单位球上均匀分布或正态分布的模型。对于定义为$\pm X_1,\ldots,\pm X_n$和$0,X_1,\ldots,X_n$凸包的随机多面体，我们得到了类似的结果.证明中使用的主要工具之一是Blaschke-Petkantschin公式。

英文摘要：

Let $X_1,\ldots,X_n$ be i.i.d.\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,β} (x) = \text{const} \cdot (1-\|x\|^2)^β, \quad \|x\|\leq 1, \quad \text{(the beta case)} $$ and $$ \tilde f_{d,β} (x) = \text{const} \cdot (1+\|x\|^2)^{-β}, \quad x\in\mathbb{R}^d, \quad \text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when $β\downarrow -1$, respectively $β\uparrow +\infty$, we can also cover the models in which $X_1,\ldots,X_n$ are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of $\pm X_1,\ldots,\pm X_n$ and $0,X_1,\ldots,X_n$. One of the main tools used in the proofs is the Blaschke-Petkantschin formula.

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