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Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes
Discrete & Computational Geometry ( IF 0.8 ) Pub Date : 2020-12-10 , DOI: 10.1007/s00454-020-00259-z
Zakhar Kabluchko

Consider a random simplex $[X_1,\ldots,X_n]$ defined as the convex hull of independent identically distributed (i.i.d.) random points $X_1,\ldots,X_n$ in $\mathbb{R}^{n-1}$ with the following beta density: $$ f_{n-1,\beta} (x) \propto (1-\|x\|^2)^{\beta} 1_{\{\|x\| < 1\}}, \qquad x\in\mathbb{R}^{n-1}. $$ Let $J_{n,k}(\beta)$ be the expected internal angle of the simplex $[X_1,\ldots,X_n]$ at its face $[X_1,\ldots,X_k]$. Define $\tilde J_{n,k}(\beta)$ analogously for i.i.d. random points distributed according to the beta' density $$ \tilde f_{n-1,\beta} (x) \propto (1+\|x\|^2)^{-\beta}, \qquad x\in\mathbb{R}^{n-1}. $$ We derive formulae for $J_{n,k}(\beta)$ and $\tilde J_{n,k}(\beta)$ which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of $\beta$. 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 explicitly 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 [Adv. Math., 2005] proved the existence of the limit of the normalized expected $f$-vector of $K_{n,d}$: $$ \lim_{n\to\infty} n^{-{\frac{d-1}{d+1}}}\mathbb E \mathbf f(K_{n,d}) = \mathbf c_d \cdot \Omega(K), $$ where $\Omega(K)$ 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.

中文翻译:

随机单纯形角的递归方案,以及对随机多面体的应用

考虑一个随机单纯形 $[X_1,\ldots,X_n]$ 定义为 $\mathbb{R}^{n-1}$ 中独立同分布(iid)随机点 $X_1,\ldots,X_n$ 的凸包具有以下 beta 密度: $$ f_{n-1,\beta} (x) \propto (1-\|x\|^2)^{\beta} 1_{\{\|x\| < 1\}},\qquad x\in\mathbb{R}^{n-1}。$$ 令 $J_{n,k}(\beta)$ 为单纯形 $[X_1,\ldots,X_n]$ 在其面 $[X_1,\ldots,X_k]$ 处的预期内角。类似地定义 $\tilde J_{n,k}(\beta)$ 根据 beta' 密度分布的 iid 随机点 $$ \tilde f_{n-1,\beta} (x) \propto (1+\| x\|^2)^{-\beta}, \qquad x\in\mathbb{R}^{n-1}。$$ 我们推导出 $J_{n,k}(\beta)$ 和 $\tilde J_{n,k}(\beta)$ 的公式,这使得可以在有限多步中象征性地计算这些量,对于任何$\beta$ 的整数或半整数值。对于 $J_{n, 1}(\pm 1/2)$ 我们甚至提供了关于 Gamma 函数乘积的明确公式。我们将这些结果应用于两个看似无关的随机几何问题。(i) 我们明确计算了典型 Poisson-Voronoi 单元的预期 $f$-vectors,维度高达 $10$。(ii) 考虑随机多面体 $K_{n,d} := [U_1,\ldots,U_n]$ 其中 $U_1,\ldots,U_n$ 是在某个 $d$ 维凸体 $ 内均匀采样的 iid 随机点 $ K$ 具有平滑边界和单位体积。M. Reitzner [Adv. Math., 2005] 证明了 $K_{n,d}$ 的归一化期望 $f$-vector 的极限的存在: $$ \lim_{n\to\infty} n^{-{\frac{d -1}{d+1}}}\mathbb E \mathbf f(K_{n,d}) = \mathbf c_d \cdot \Omega(K), $$ 其中 $\Omega(K)$ 是仿射面$K$ 的面积,$\mathbf c_d$ 是一个不依赖于 $K$ 的未知向量。
更新日期:2020-12-10
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