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**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.

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.

Reviewer: Andrei Sipoş (Darmstadt)

### MSC:

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |

54C65 | Selections in general topology |

### Keywords:

Chebyshev set; sun; strict sun; radial continuity; bounded solarity; selection of the metric projection operator; best approximation; near-best approximation
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\textit{A. R. Alimov}, Set-Valued Var. Anal. 27, No. 1, 213--222 (2019; Zbl 1423.41050)

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