Strong proximinality in metric spaces

Sahil Gupta, T. D. Narang

In this paper, we extend the notions of strong proximinality and strong Chebyshevity available in Banach spaces to metric spaces and prove that an approximatively compact subset $W$ of a metric space $X$ is strongly proximinal. Moreover, the converse holds if the set of best approximants in $W$ to each point of the space $X$ is compact. It is proved that strongly Chebyshev sets are precisely the sets which are strongly proximinal and Chebyshev. Further, by extending the notion of local uniform convexity from Banach spaces to metric spaces, it is proved that a proximinal convex subset of a locally uniformly convex metric space is approximatively compact. As a consequence, it is observed that closed balls in a locally uniformly convex metric space are strongly Chebyshev.