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Selections of the metric projection operator and strict solarity of sets with continuous metric projection. (English. Russian original) Zbl 1426.41039
Sb. Math. 208, No. 7, 915-928 (2017); translation from Mat. Sb. 208, No. 7, 3-18 (2017).
For a normed linear space \(X\) and a nonempty subset \(M\), the metric projection \(P_M\) is a set-valued mapping of \(X\) to \(M\) defined by \(P_Mx=\{y \in M\mid \inf _{z \in M}\|x-z\| =\|x-y\|\}\) for \(x \in X\). A selection of the metric projection is a single-valued mapping \(f : X \to M\) satisfying \(f(x) \in P_Mx\) for any \(x \in X\). The set \(M\) is called a sun if for any \(x \in X \setminus M\) there exists \(y \in P_Mx\) such that \(y \in P_M((1-\lambda)y+\lambda x)\) for all \(\lambda\geq0\), and \(M\) is called an existence set if \(P_Mx \ne \varnothing\) for any \(x \in X\).
I. G. Tsar’kov [Math. Notes 47, No. 2, 218–227 (1990; Zbl 0703.46010); translation from Mat. Zametki 47, No. 2, 137–148 (1990)] proved that in a finite-dimensional Banach space an existence set with lower semicontinuous metric projection is a sun and has acyclic intersections with closed balls. In this paper, the author proves the following theorem: Let \(X\) be a finite-dimensional Banach space with one of the following properties: (1) \(\dim X\leq 3\); (2) \(X\) is a (BM)-space introduced by A. L. Brown [Math. Ann. 279, No. 1–2, 87–101 (1987; Zbl 0607.41027)]; (3) \(X\) is in the class (RBR) introduced by N. V. Nevesenko [Math. Notes 23, 308–312 (1978; Zbl 0408.41019)]. Then any closed set with lower semicontinuous metric projection in \(X\) is a sun, admits a continuous selection of the metric projection, has contractible intersections with balls, and its (nonempty) intersection with any closed ball is a retract of this ball. The author also proves theorems on related properties.

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