Extreme values for characteristic radii of a Poisson-Voronoi Tessellation
Extreme values for characteristic radii of a Poisson-Voronoi Tessellation
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Poisson-Voronoi 曲面细分的特征半径的极值
作者：
P. Calka;Nicolas Chenavier
在凸体 W 中观察到强度为 γ 的均匀 Poisson-Voronoi 曲面细分。我们将曲面细分的每个单元与两个特征半径相关联：内半径，即以核为中心并包含在单元中的最大球的半径，以及外接半径，即以核为中心并包含单元的最小球的半径。我们研究了 W 中所有细胞核的这两个半径的最大值和最小值。我们证明，当 γ→∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$gamma
ightarrow infty $end{document}，这四个量收敛到 Gumbel 或 Weibull 分布直至重新缩放。此外，边界单元的贡献可以忽略不计。这种方法的动机是对镶嵌的全局规律性的分析。特别是，我们研究的结果包括收敛到具有最小外接半径的单元的单纯形形状以及 W 与其所谓的 Poisson-Voronoi 近似之间的 Hausdorff 距离的上限。
A homogeneous Poisson-Voronoi tessellation of intensity γ is observed in a convex body W. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in W. We prove that when γ→∞documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$gamma
ightarrow infty $end{document}, these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between W and its so-called Poisson-Voronoi approximation.
平稳 Voronoi 曲面细分中节点点过程的二阶性质
影响因子：
1
作者：
Heinrich
通讯作者：
Heinrich