On proximinality of convex sets in superspaces |
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Authors: | Li Xin Cheng Zheng Hua Luo Wen Zhang Ben Tuo Zheng |
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Institution: | 1.School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China;2.School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, P. R. China;3.Department of Mathematical Sciences, the University of Memphis, Memphis, TN 38152-3240, USA |
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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. |
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Keywords: | Proximinality convex set local compactness Banach space |
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