Random polytopes and the Efron--Stein jackknife inequality
Random polytopes and the Efron--Stein jackknife inequality
复制标题
随机多胞形和埃夫隆-斯坦刀刀不等式
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
M. Reitzner
中科院分区：
文献类型：
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作者：
M. Reitzner
设K为光滑凸体。K中独立随机点的凸包是一个随机多面体。得到了随机多面体的体积方差和顶点数方差的估计。关键的一步是利用Efron-Stein折刀不等式求对称统计量的方差。结果是随机多面体的体积和顶点数量的大数定律。证实了Barany关于凸体的随机和最佳逼近的一个猜想。给出了凸体边界上有顶点的随机多面体的类似结果。
Let K be a smooth convex body. The convex hull of independent random points in K is a random polytope. Estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained. The essential step is the use of the Efron-Stein jackknife inequality for the variance of symmetric statistics. Consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope. A conjecture of Barany concerning random and best-approximation of convex bodies is confirmed. Analogous results for random polytopes with vertices on the boundary of the convex body are given.