On the sum of simultaneously proximinal sets

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

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