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.


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