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.

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.

Reviewer: Takamitsu Yamauchi (Matsuyama)

##### MSC:

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

54C65 | Selections in general topology |

##### Keywords:

monotone path-connected set; strict sun; sun; selection of metric projection; lower semicontinuous metric projection
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\textit{A. R. Alimov}, Sb. Math. 208, No. 7, 915--928 (2017; Zbl 1426.41039); translation from Mat. Sb. 208, No. 7, 3--18 (2017)

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##### References:

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