Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 22 (2015), No. 3, 673--685Copyright Heldermann Verlag 2015 Ball Proximinal and Strongly Ball Proximinal Spaces Pei-Kee Lin Dept. of Mathematics, University of Memphis, Memphis, TN 38152, U.S.A. pklin@memphis.edu Wen Zhang School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China wenzhang@xmu.edu.cn Bentuo Zheng Dept. of Mathematics, University of Memphis, Memphis, TN 38152, U.S.A. bzheng@memphis.edu [Abstract-pdf] 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]. Keywords: Ball proximinal, strongly ball proximinal, E-proximinal. MSC: 46B20, 41A50 [ Fulltext-pdf  (131  KB)] for subscribers only.