On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

被引：0
作者：
Dariusz Zagrodny
机构：
[1] Cardinal Stefan Wyszyński University,Faculty of Mathematics and Natural Sciences, College of Science
来源：
Set-Valued and Variational Analysis | 2015年 / 23卷
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;
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学科分类号：
摘要：
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.
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页码：581 / 612
页数：31
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