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Convexity and the continuity of metric projections. (Chinese. English summary) Zbl 1134.41338

Summary: We study the continuity of metric projections in nearly strong convex, nearly very convex Banach spaces. The following results are obtained: Let \(A\) be a proximinal convex subset in a nearly strong convex (nearly very convex) Banach space; then the metric projection \(P_ A\) is norm-norm upper semicontinuous (norm-weak upper semicontinuous). Moreover, using the continuity of metric projections, we give some characterizations of nearly strong convexity and nearly very convexity.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20 Geometry and structure of normed linear spaces
49K40 Sensitivity, stability, well-posedness
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