Random polytopes and the Efron-Stein jackknife inequality

• 成果类型: Article
• 署名作者: Reitzner, M
• 署名单位: Technische Universitat Wien; University of Freiburg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1068646381
• 发表日期: 2003
• 页码: 2136-2166
• 关键词: smooth convex-bodies; LIMIT-THEOREMS; stepwise approximation; random points; mean-value; body; set; hulls; plane; triangulation

摘要

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.