This is an announcement for the paper “On the sum of simultaneously proximinal sets” by Longfa Sunhttps://arxiv.org/search/math?searchtype=author&query=Sun%2C+L, Yuqi Sunhttps://arxiv.org/search/math?searchtype=author&query=Sun%2C+Y, Wen Zhanghttps://arxiv.org/search/math?searchtype=author&query=Zhang%2C+W, Zheming Zhenghttps://arxiv.org/search/math?searchtype=author&query=Zheng%2C+Z.

Abstract: 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.

https://arxiv.org/abs/2002.11961