Convexities and approximative compactness and continuity of metric projection in Banach spaces. (English) Zbl 1190.46018

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.


46B20 Geometry and structure of normed linear spaces
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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