Abstract

Two max- and min-approximation problems on solarity of sets in dual spaces are considered. It is shown that if the metric projection onto a set \(M\subset X^{*}\) is \(w^{*}\)-upper semicontinuous and has nonempty \(w^{*}\)-closed acyclic values, then \(M\) is a sun. In particular, a Chebyshev set with \(w^{*}\)-continuous metric projection is a sun. In the max-approximation setting, a set with \(w^{*}\)-upper-semicontinuous \(\max\)-projection with nonempty \(w^{*}\)-closed acyclic values is shown to be local \(\max\)-sun. As a result, it follows that that a uniquely remotal set with \(w^{*}\)-continuous \(\max\)-projection operator is a singleton, which gives an answer to the well-known unique farthest point problem in dual spaces for sets with \(w^{*}\)-continuous farthest-point mapping.

中文翻译：

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Chebyshev 集在对偶空间和唯一远程集的太阳度

摘要

考虑了对偶空间中集合的日光度的两个最大和最小逼近问题。结果表明，如果在集合\(M\subset X^{*}\)上的度量投影是\(w^{*}\) -上半连续的并且具有非空\(w^{*}\) -closed非循环值，则\(M\)是一个太阳。特别是，具有\(w^{*}\) - 连续度量投影的切比雪夫集是一个太阳。在最大近似设置，一组具有\（W ^ {*} \） -上-半\（\最大\） -projection与非空\（W ^ {*} \） -封闭无环的值被示出为本地\(\max\) -sun. 因此，一个唯一的远程集具有\(w^{*}\) -continuous \(\max\) -projection 算子是一个单例，它给出了对偶空间中具有\(w^{*} 的集合的众所周知的唯一最远点问题的答案\) - 连续最远点映射。