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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:
46B20Geometry and structure of normed linear spaces
41A65Abstract approximation theory
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References:
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