Relations between certain classes of sets in Banach spaces (Q579578)

Let X be a real normed linear space and M a non-empty subset of X. For \(x\in X\) write \(P_ Mx=\{y\in M:\| x-y\| =d(x,M)\}\), where \(d(x,M)=\inf \{\| x-m\|:m\in M\}\). M is called approximately compact if for each sequence \(\{y_ n\}\) in M such that \(\| x-y_ n\| \to d(x,M)\) there exists a subsequence converging in M. Denote by \({\mathcal Z}\) the set of all non-empty closed subsets of X and by \((P_ 0)\) (respectively (P)) the family of subsets M of X for which \(P_ Mx\) is connected (respectively connected and non-empty) for all x. The following results are typical for the theorems proved in the paper: (1) Let X be a Banach space with the property that each hyperplane is approximately compact. Then each \(M\in (P_ 0)\cap {\mathcal Z}\) has the property that \(M\cap \{y:\| y-x\| <r\}\) is connected for all x and all \(r>0.\) (2) \((P_ 0)\cap {\mathcal Z}=(P)\) if and only if X is finite-dimensional.