Asymptotic Shapes of large Cells in Random Tessellations
Asymptotic Shapes of large Cells in Random Tessellations
复制标题
随机镶嵌中大细胞的渐近形状
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
R. Schneider
中科院分区：
文献类型：
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作者：
D. Hug;R. Schneider
翻译后摘要：我们建立了一个紧密的关系，等周不等式凸体和渐近形状的大型随机多面体，这是细胞在某些随机镶嵌在d维欧几里德空间。这些马赛克生成的泊松超平面过程满足一些自然的假设（不一定平稳或各向同性）。大胞腔的大小由一类一般泛函来度量。主要结果意味着大胞腔的渐近形状完全由等周型不等式的极值体决定，该不等式将尺寸泛函与生成过程的超平面的期望数目联系起来。在细胞具有大尺寸的条件下，我们得到了零细胞从渐近或极限形状的大偏差的条件概率的指数估计。
Abstract.We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells from asymptotic or limit shapes, under the condition that the cells have large size.