Properties of sets that have a continuous selection of the operator \(P^{\delta}\)
If X is a Banach space, \(M\subset X\), \(x\in X\) and \(\delta >0\), then \(P_ M^{\delta}(x)=\{y\in M;\quad \| x-y\| \leq dist(x,M)+\delta \}.\) The interplay of the following properties a), b), c) is investigated: a) the existence of continuous selection for the operator \(x\to P_ M^{\delta}(x),\) b) approximate compactness of M, c) convexity properties of M. Typical result (Corollary 2). If \(X^*\) is strictly convex, \(M\subset X\) is approximately compact and \(P_ M^{\delta}\) admits a continuous selection for every \(\delta >0\), then M is convex.
scientific article; zbMATH DE number 4205137
scientific article