Porosity of mutually nearest and mutually furthest points in Banach spaces
Porosity of mutually nearest and mutually furthest points in Banach spaces
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DOI：
10.1016/j.jat.2003.07.001
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发表时间：
2003-11
期刊：
影响因子：
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通讯作者：
Chong Li;Hong-Kun Xu
中科院分区：
文献类型：
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作者：
Chong Li;Hong-Kun Xu
设X是真实的严格凸Kadec Banach空间，G是X的非空闭相对有界弱紧子集.设B（X）（分别K（X））是非空有界闭（或设BG（X）表示集合{A∈ B（X）：A <$G=<$K}的闭包，KG（X）= BG（X）<$K（X）.本文引入了B（X）的容许族A，并证明了EAo（G）（resp. EoA（G）），所有子集F∈ A <$BG（X）的集合（分别F∈ A B（X））使得极小化问题min（F，G）（resp.极大化问题max（F，G））是适定的，是A的一个稠密Gδ-子集。当X是一致凸的时，我们证明了A <$EoA（G）和A <$EoA（G）在A中是σ-多孔的.
Let X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X. Let B (X) (resp. K (X) ) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let BG(X) denote the closure of the set {A∈ B (X) : A∩G=∅} and KG(X)= BG(X)∩ K (X) . We introduce the admissible family A of B (X) and prove that EAo(G) (resp. EoA(G) ), the set of all subsets F∈ A ⊆ BG(X) (resp. F∈ A ⊆ B (X) ) such that the minimization problem min(F,G) (resp. the maximization problem max(F,G)) is well-posed, is a dense Gδ-subset of A . Furthermore, when X is uniformly convex, we prove that A ⧹EAo(G) and A ⧹EoA(G) are σ-porous in A .