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.
