On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

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• 作者：Dariusz Zagrodny
• 关键词：Inverse mapping of metric projection ; Chebyshev sets ; Best approximation ; Convexity ; Differentiability of the distance function ; Concavity of the distance function ; Primary 49J52 ; 41A50 ; Secondary 41A65 ; 52A40 ; 46C05
• 刊名：Set-Valued and Variational Analysis
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：23
• 期：4
• 页码：581-612
• 全文大小：495 KB
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• 作者单位：Dariusz Zagrodny (1)
  
  1. Faculty of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszy艅ski University, Dewajtis 5, Warsaw, Poland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1877-0541

文摘

The closure of preimages (inverse images) of metric projection mappings to a given set in a Hilbert space are investigated. In particular, some properties of fibers over singletons (level sets or preimages of singletons) of the metric projection are provided. One of them, a sufficient condition for the convergence of minimizing sequence for a giving point, ensures the convergence of a subsequence of minimizing points, thus the limit of the subsequence belongs to the image of the metric projection. Several examples preserving this sufficient condition are provided. It is also shown that the set of points for which the sufficient condition can be applied is dense in the boundary of the preimage of each set from a large class of subsets of the Hilbert space. As an application of obtained properties of preimages we show that if the complement of a nonconvex set is a countable union of preimages of convex closed sets then there is a point such that the value of the metric projection mapping is not a singleton. It is also shown that the Klee result, stating that only convex closed sets can be weakly closed Chebyshev sets, can be obtained for locally weakly closed sets. Keywords Inverse mapping of metric projection Chebyshev sets Best approximation Convexity Differentiability of the distance function Concavity of the distance function