On proximinality of convex sets in superspaces
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.
Acta Mathematica Sinica, English Series
Cheng, L., Luo, Z., Zhang, W., & Zheng, B. (2016). On proximinality of convex sets in superspaces. Acta Mathematica Sinica, English Series, 32 (6), 633-642. https://doi.org/10.1007/s10114-016-5355-0