Proximinality in Banach spaces (Q2474938)

Let \(X\) be a Banach space and let \(\tau\) be the norm or the weak topology of \(X.\) The paper is concerned with various \(\tau\)-notions related to the approximation properties of \(\tau\)-closed subsets of \(X\). For instance, a \(\tau\)-closed subset \(K\) of \(X\) is called \(\tau\)-strongly proximinal if, for every \(\tau\)-neighborhood \(V\) of \(0\in X\), there exists \(\delta>0\) such that \(P_K(x,\delta)\subset P_K(x)+V\), where \(P_K(x)\) is the metric projection of \(x\) on \(K\) and \(P_K(x,\delta)=\{z\in K:\| x-z\|\leq d(x,K)+\delta\}\). The set \(K\) is called \(\tau\)-strongly Chebyshev (approximatively \(\tau\)-compact) if, for any \(x\in X\setminus K\), every minimizing sequence for \(d(x,K)\) is \(\tau\)-convergent (contains a \(\tau\)-convergent subsequence). Strongly proximinal subspaces, in the case when \(\tau\) is the norm-topology, were defined and studied by \textit{G.\,Godefroy} and \textit{V.\,Indumathi} [Rev.\ Mat.\ Complut.\ 14, No.\,1, 105--125 (2001; Zbl 0993.46004)]. The authors study the relations of these notions with the geometric properties of the Banach space \(X\) (as, for instance, the \(\tau\)-almost local rotundity, a notion considered by \textit{P.\,Bandyopadhyay, D.\,Huang, B.--L.\thinspace Lin} and \textit{S.\,L.\thinspace Troyanski} [J.~Math.\ Anal.\ Appl.\ 252, No.\,2, 906--916 (2000; Zbl 0978.46004)]) and with various continuity properties of the metric projection. For instance, for \(x^*\in S_{X^*},\) the subspace \(\ker x^*\) is approximatively \(\tau\)-compact (respectively, \(\tau\)-strongly Chebyshev) iff \(x^*\) is a \(\tau\)-strongly support (respectively, a \(\tau\)-strongly exposing) functional of the unit ball of \(X\) (Theorem~2.10). The paper ends with some stability results for \(\ell^p\)-sums of Banach spaces.