Approximative properties of sets and continuous selections. (English. Russian original) Zbl 1455.54019
Sb. Math. 211, No. 8, 1190-1211 (2020); translation from Mat. Sb. 211, No. 8, 132-157 (2020).
The celebrated Michael continuous selection theorem is extended to metric projections. The following result is proved:
Theorem 1. Let \((X,\|\cdot\|)\) be an asymmetric finite-dimensional space and suppose that \(M\subset X.\) If the metric projection onto \(M\) is lower semicontinuous, then it has a continuous selection, that is, there exists a continuous mapping \(\varphi\colon X\to M\) such that \(\varphi(x)=P_Mx\) for all \(x\in X.\)
Continuous selections of multifunctions of certain types are studied (solarity, suns and so on). It is also proved that the relative Chebyshev-centre map admits a continuous selection.
Theorem 1. Let \((X,\|\cdot\|)\) be an asymmetric finite-dimensional space and suppose that \(M\subset X.\) If the metric projection onto \(M\) is lower semicontinuous, then it has a continuous selection, that is, there exists a continuous mapping \(\varphi\colon X\to M\) such that \(\varphi(x)=P_Mx\) for all \(x\in X.\)
Continuous selections of multifunctions of certain types are studied (solarity, suns and so on). It is also proved that the relative Chebyshev-centre map admits a continuous selection.
Reviewer: Szymon Wąsowicz (Bielsko-Biała)
MSC:
| 54C60 | Set-valued maps in general topology |
| 41A28 | Simultaneous approximation |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 47A52 | Linear operators and ill-posed problems, regularization |
| 46B20 | Geometry and structure of normed linear spaces |
| 54C65 | Selections in general topology |
Keywords:
set-valued mapping; continuous selection; sun; monotone path-connected set; relative Chebyshev centre and pointReferences:
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