Simultaneous strong proximinality in Banach spaces

Several researchers have discussed the problem of strong proximinality in Banach spaces. In this paper, we generalize the notion of strong proximinality and define simultaneous strong proximinality. It is proved that if $W$ is a simultaneously approximatively compact subset of a Banach space $X$ then $W$ is simultaneously strongly proximinal and the converse holds if the set of all best simultaneous approximations to every bounded subset $S$ of $X$ from $W$ is compact. We show that simultaneously strongly Chebyshev sets are precisely the sets that are simultaneously strongly proximinal and simultaneously Chebyshev. It is also proved that if $F$ and $W$ are subspaces of a Banach space $X$ such that $F$ is simultaneously strongly proximinal, $W$ is finite dimensional and $F+W$ is closed then $F+W$ is simultaneously strongly proximinal in $X$. How simultaneous strong proximinality is transmitted to and from quotient spaces is discussed in this paper.