## 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