Convexities and approximative compactness and continuity of metric projection in Banach spaces (Q1048990)

The paper is concerned with the relations between various geometric properties of a Banach space \(X\) -- strong convexity, nearly strong convexity, the property of being very convex or nearly very convex -- and the continuity properties of the metric projection, as upper semi-continuity (usc), strong Wijsman-Zhang usc, on closed convex subsets of \(X.\) The relevance of the approximative compactness for the continuity of the metric projection is also emphasized. For instance, in a nearly strongly convex Banach space, a closed convex subset \(A\) of \(X\) is proximinal with usc metric projection \(P_A\) iff \(A\) is approximatively compact. The paper also contains a representation of the metric projection on \(w^*\)-closed hyperplanes in \(X^*\) and conditions ensuring its upper semi-continuity. The obtained results improve some previous results obtained by several authors.