  In this paper, we show that the sum of a compact convex subset and a
simultaneously $\tau$-strongly proximinal convex subset (resp. simultaneously
approximatively $\tau$-compact convex subset) of a Banach space X is
simultaneously tau-strongly proximinal (resp. simultaneously approximatively
$\tau$-compact ), and the sum of weakly compact convex subset and a
simultaneously approximatively weakly compact convex subset of X is still
simultaneously approximatively weakly compact, where $\tau$ is the norm or the
weak topology. Moreover, some related results on the sum of simultaneously
proximinal subspaces are presented.
