## Continuity of the metric projection and local solar properties of sets. Continuity of the metric projection and local solar properties of sets, continuity of the metric projection and solar properties.(English)Zbl 1423.41050

Let $$M$$ be a nonempty closed subset of a real normed space $$X$$ and $$P_M$$ be the (set-valued) metric projection on $$M$$. The set $$M$$ is called a sun if for every $$x \in X \setminus M$$ there is a $$y \in P_M(x)$$ such that for all $$\lambda \geq 0$$, $$y \in P_M(y+\lambda (y-x))$$. If in addition, for any $$x \in X \setminus M$$, $$P_M(x) \neq\emptyset$$, and the condition from before holds for any $$y \in P_M(x)$$, then $$M$$ is called a strict sun. $$M$$ is called an LG-set if for any $$x \notin M$$, $$\varepsilon > 0$$ and $$y \in P_{M \cap B(y,\varepsilon)}$$, $$y \in P_M(x)$$.
Let $$Q$$ be a property that subsets of $$X$$ can have. Then we say that $$M$$ has property $$P$$-$$Q$$ if for all $$x \in X$$, $$P_M(x)$$ is nonempty and has property $$Q$$ and that it has property $$B$$-$$Q$$ if for all $$x \in X$$ and $$r > 0$$, $$M \cap B(x,r)$$ has property $$Q$$ or is empty.
The author obtains some implications and equivalences among these and similar notions, e.g. that any LG-set that is a $$B$$-sun is a strict sun and that a $$P$$-compact $$B$$-sun is a sun. In addition, he derives sufficient conditions for a closed locally Chebyshev set to be Chebyshev.

### MSC:

 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 54C65 Selections in general topology
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