On Proximinality of Convex Sets in Superspaces  

作　　者：Li Xin CHENG Zheng Hua LUO Wen ZHANG Ben Tuo ZHENG 

机构地区：[1]School of Mathematical Sciences, Xiamen University [2]School of Mathematical Sciences, Huaqiao University [3]Department of Mathematical Sciences, the University of Memphis

出　　处：《Acta Mathematica Sinica,English Series》2016年第6期633-642,共10页数学学报（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11371296);supported by National Natural Science Foundation of China(Grant No.11201160);supported by National Natural Science Foundation of China(Grant No.11471270);Ph.D Programs Foundation of MEC(Grant No.20130121110032);Natural Science Foundation of Fujian Province(Grant No.2012J05006);Natural Science Foundation of Fujian Province(Grant No.2015J01022);supported by NSF(Grant No.DMS-1200370)

摘　　要：Abstract In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal （proximinal, resp.） in every Banach space isometrically containing it if and only if C is locally （weakly, resp.） compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.Abstract In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal （proximinal, resp.） in every Banach space isometrically containing it if and only if C is locally （weakly, resp.） compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.

关 键 词：Proximinality convex set local compactness Banach space 

分 类 号：O177.2[理学—数学]