We study random polytopes of the form $[X_1,\ldots,X_n]$ defined as convex hulls of independent identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^d$ with one of the following densities: $$ f_{d,\beta} (x) = c_{d,\beta} (1-\|x\|^2)^{\beta}, \qquad \|x\| < 1, \quad \text{(beta distribution)} $$ or $$ \tilde f_{d,\beta} (x) = \tilde{c}_{d,\beta} (1+\|x\|^2)^{-\beta}, \qquad x\in\mathbb{R}^d, \quad \text{(beta' distribution)}. $$ This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of $k$-faces of $[X_1,\ldots,X_n]$ for arbitrary $k\in\{0,1,\ldots,d-1\}$. We prove that for any such $k$ this expected number is strictly monotonically increasing with $n$. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large $n$ limit in the beta' case, we compute the expected $f$-vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of $k$-faces of the zero cell for a class of isotropic Poisson hyperplane tessellation in $\mathbb{R}^d$. This family includes the zero cell of a classical stationary and an isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson--Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this $f$-vector in the high-dimensional regime, as $d\to\infty$. Finally, we relate the $d$-dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory.

中文翻译：

---

Beta 多面体和泊松多面体：f 向量和角度

我们研究了 $[X_1,\ldots,X_n]$ 形式的随机多胞体，定义为 $\mathbb{R}^d$ 中独立同分布随机点 $X_1,\ldots,X_n$ 的凸包，具有以下其中一项密度：$$ f_{d,\beta} (x) = c_{d,\beta} (1-\|x\|^2)^{\beta}, \qquad \|x\| < 1, \quad \text{(beta 分布)} $$ 或 $$ \tilde f_{d,\beta} (x) = \tilde{c}_{d,\beta} (1+\|x\ |^2)^{-\beta}, \qquad x\in\mathbb{R}^d, \quad \text{(beta' distribution)}。$$ 此设置还包括单位球体上的均匀分布和作为极限情况的标准正态分布。我们为任意 $k\in\{0,1,\ldots,d-1\}$ 的 $[X_1,\ldots,X_n]$ 的 $k$-faces 的预期数量推导出精确和渐近公式。我们证明，对于任何这样的 $k$，这个预期数字严格地随着 $n$ 单调增加。还，我们计算这些多胞体在每个维度上的预期内角和外角，更一般地说，计算它们的切锥的预期圆锥内在体积。通过传递到 beta' 情况下的大 $n$ 限制，我们计算具有幂律强度函数的 Poisson 点过程的凸包的预期 $f$ 向量。使用凸对偶性，我们推导出了 $\mathbb{R}^d$ 中一类各向同性泊松超平面细分的零单元的 $k$-面的预期数量的精确公式。该族包括经典平稳和各向同性泊松超平面细分的零单元以及作为特殊情况的平稳泊松-Voronoi 细分的典型单元。此外，我们证明了这个 $f$-vector 在高维状态下的精确极限定理，如 $d\to\infty$。最后，