Full length article: Strong proximinality of closed convex sets
Full length article: Strong proximinality of closed convex sets
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DOI：
10.1016/j.jat.2011.01.001
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发表时间：
2011-04
影响因子：
0.9
通讯作者：
S. Dutta;P. Shunmugaraj
中科院分区：
文献类型：
--
作者：
S. Dutta;P. Shunmugaraj
我们证明了在Banach空间X中，每个闭凸子集是强逼近的当且仅当对偶范数是强次可微的，并且对于对偶空间X中的每个范数为1的泛函f，X中的范数为1的元素的集合JX（f）是紧的，其中f达到它的范数.因此，我们观察到，如果对偶范数是强次可微的，则X的每个闭凸子集是强逼近的当且仅当X的每个闭凸子集上的度量投影是上半连续的。
We show that in a Banach space X, every closed convex subset is strongly proximinal if and only if the dual norm is strongly subdifferentiable and for each norm 1 functional f in the dual space X∗, JX(f)- the set of norm 1 elements in X where f attains its norm - is compact. As a consequence, it is observed that if the dual norm is strongly subdifferentiable, then every closed convex subset of X is strongly proximinal if and only if the metric projection onto every closed convex subset of X is upper semi-continuous.