On the sum of simultaneously proximinal sets

In this paper, we show that the sum of a compact convex subset and a simultaneously $\tau$-strongly proximinal convex subset insert ignore into journalissuearticles values(resp. simultaneously approximatively $\tau$-compact convex subset); of a Banach space X is simultaneously $\tau$-strongly proximinal insert ignore into journalissuearticles values(resp. simultaneously approximatively $\tau$-compact );, and the sum of a 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.