Title: Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability Show Chinese title

Title: Beta 多面体和 Beta 锥：几何概率中的一个精确可解模型

Abstract: Let $X_1,\ldots, X_n$ be independent random points in the unit ball of $\mathbb R^d$ such that $X_i$ follows a beta distribution with the density proportional to $(1-\|x\|^2)^{\beta_i}1_{\{\|x\| <1\}}$. Here, $\beta_1,\ldots, \beta_n> -1$ are parameters. We study random polytopes of the form $[X_1,\ldots,X_n]$, called beta polytopes. We determine explicitly expected values of several functionals of these polytopes including the number of $k$-dimensional faces, the volume, the intrinsic volumes, the total $k$-volume of the $k$-skeleton, various angle sums, and the $S$-functional which generalizes and unifies many of the above examples. We identify and study the central object needed to analyze beta polytopes: beta cones. For these, we determine explicitly expected values of several functionals including the solid angle, conic intrinsic volumes and the number of $k$-dimensional faces. We identify expected conic intrinsic volumes of beta cones as a crucial quantity needed to express all the functionals mentioned above. We obtain a formula for these expected conic intrinsic volumes in terms of a function $\Theta$ for which we provide an explicit integral representation. The proofs combine methods from integral and stochastic geometry with the study of the analytic properties of the function $\Theta$. Show Chinese abstract

Abstract: Let $X_1,\ldots, X_n$ be independent random points in the unit ball of $\mathbb R^d$ such that $X_i$ follows a beta distribution with the density proportional to $(1-\|x\|^2)^{\beta_i}1_{\{\|x\| <1\}}$. Here, $\beta_1,\ldots, \beta_n> -1$ are parameters. We study random polytopes of the form $[X_1,\ldots,X_n]$, called beta polytopes. 我们明确确定了这些多面体的几个泛函的期望值，包括$k$维面的数量、体积、内禀体积、$k$维骨架的总$k$体积、各种角度和，以及$S$泛函，该泛函推广并统一了许多上述例子。 我们识别并研究了分析beta多面体所需的核心对象：beta锥体。 对于这些对象，我们明确确定了几个泛函的期望值，包括立体角、圆锥内禀体积和$k$维面的数量。 我们将beta锥体的期望圆锥内禀体积识别为表达上述所有泛函的关键量。 我们得到了这些期望圆锥内禀体积的公式，该公式以函数$\Theta$表示，我们为其提供了显式的积分表示。 证明结合了积分和随机几何的方法以及对函数$\Theta$的解析性质的研究。