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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.

MSC:

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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