随机单形角的递归格式及其在随机多面体中的应用
Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes
摘要：
考虑一个随机单形$[X_1,\ldots,X_n]$，定义为$\mathbb{R}^{n-1}$中独立同分布的随机点$X_1,\ldots,X_n$的凸包，具有以下β密度：$$ f_{n-1,β} (x) \propto (1-\|x\|^2)^β 1_{\{\|x\| < 1\}}, \qquad x\in\mathbb{R}^{n-1}, \quad β>-1. $$令$J_{n,k}(β)$为单形$[X_1,\ldots,X_n]$在其面$[X_1,\ldots,X_k]$的期望内角。与i.i.d.类似地定义$\tilde J_{n,k}(β)$。根据β '密度分布的随机点$$ \tilde f_{n-1,β} (x) \propto (1+\|x\|^2)^{-β}, \qquad x\in\mathbb{R}^{n-1}, \quad β> \frac{n-1}{2}. $$我们推导出$J_{n,k}(β)$和$\tilde J_{n,k}(β)$的公式，这些公式使得对于$β$的任何整数或半整数值，可以符号化地、在许多步骤中计算这些量。对于$J_{n,1}(\pm 1/2)$，我们甚至提供了明确的公式，在产品的伽玛功能。我们将这些结果应用于两个看似无关的随机几何问题。(i)我们计算预期的$f$向量的典型泊松-Voronoi细胞的尺寸$10$。(ii)考虑随机多面体$K_{n,d} := [U_1,\ldots,U_n]$，其中$U_1,\ldots,U_n$是i.i.d.。$d$维凸体$K$内均匀采样的随机点，具有光滑边界和单位体积。M. Reitzner证明了$K_{n,d}$的归一化期望$f$ -向量的极限的存在：$$ \lim_{n\to\infty} n^{-{\frac{d-1}{d+1}}}\mathbb E \mathbf f(K_{n,d}) = \mathbf c_d \cdot Ω(K), $$其中$Ω(K)$是$K$的仿射表面积，$\mathbf c_d$是不依赖于$K$的未知向量。我们计算$\mathbf c_d$明确的$d=10$的维度，也解决了类似的问题，随机多面体的顶点均匀分布在球体上。
英文摘要：
Consider a random simplex $[X_1,\ldots,X_n]$ defined as the convex hull of independent identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^{n-1}$ with the following beta density: $$ f_{n-1,β} (x) \propto (1-\|x\|^2)^β 1_{\{\|x\| < 1\}}, \qquad x\in\mathbb{R}^{n-1}, \quad β>-1. $$ Let $J_{n,k}(β)$ be the expected internal angle of the simplex $[X_1,\ldots,X_n]$ at its face $[X_1,\ldots,X_k]$. Define $\tilde J_{n,k}(β)$ analogously for i.i.d. random points distributed according to the beta' density $$ \tilde f_{n-1,β} (x) \propto (1+\|x\|^2)^{-β}, \qquad x\in\mathbb{R}^{n-1}, \quad β> \frac{n-1}{2}. $$ We derive formulae for $J_{n,k}(β)$ and $\tilde J_{n,k}(β)$ which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of $β$. For $J_{n,1}(\pm 1/2)$ we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry. (i) We compute the expected $f$-vectors of the typical Poisson-Voronoi cells in dimensions up to $10$. (ii) Consider the random polytope $K_{n,d} := [U_1,\ldots,U_n]$ where $U_1,\ldots,U_n$ are i.i.d. random points sampled uniformly inside some $d$-dimensional convex body $K$ with smooth boundary and unit volume. M. Reitzner proved the existence of the limit of the normalized expected $f$-vector of $K_{n,d}$: $$ \lim_{n\to\infty} n^{-{\frac{d-1}{d+1}}}\mathbb E \mathbf f(K_{n,d}) = \mathbf c_d \cdot Ω(K), $$ where $Ω(K)$ is the affine surface area of $K$, and $\mathbf c_d$ is an unknown vector not depending on $K$. We compute $\mathbf c_d$ explicitly in dimensions up to $d=10$ and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
