Geodesic
Ball Proximinal and Strongly Ball Proximinal Spaces
Journal of convex analysis, Tome 22 (2015) no. 3, pp. 673-685
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Let $Y$ be an $E$-proximinal (respectively, a strongly proximinal) subspace of $X$. We prove that $Y$ is (strongly) ball proximinal in $X$ if and only if for any $x\in X$ with $(x+Y)\cap B_X\ne\emptyset$, $(x+Y)\cap B_X$ is (strongly) proximinal in $x+Y$. Using this characterization and a smart construction, we obtain three Banach spaces $Z\subset Y\subset X$ such that $Z$ is ball proximinal in $X$ and $Y/Z$ is ball proximinal in $X/Z$, but $Y$ is not ball proximinal in $X$. This solves a problem raised by P. Bandyopadhyay, Bor-Luh Lin and T.S.S.R.K. Rao [{\em Ball proximinality in Banach spaces,} in: Banach Spaces and Their Applications in Analysis (Oxford/USA, 2006) B. Randrianantoanina et al (eds.) Proceedings in Mathematics, de Gruyter, Berlin (2007) 251--264].
Classification : 46B20, 41A50
Mots-clés : Ball proximinal, strongly ball proximinal, E-proximinal 
@article{JCA_2015_22_3_JCA_2015_22_3_a4,
     author = {P.-K. Lin and W. Zhang and B. Zheng},
     title = {Ball {Proximinal} and {Strongly} {Ball} {Proximinal} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {673--685},
     year = {2015},
     volume = {22},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2015_22_3_JCA_2015_22_3_a4/}
}
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P.-K. Lin; W. Zhang; B. Zheng. Ball Proximinal and Strongly Ball Proximinal Spaces. Journal of convex analysis, Tome 22 (2015) no. 3, pp. 673-685. http://geodesic.mathdoc.fr/item/JCA_2015_22_3_JCA_2015_22_3_a4/