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On the continuity of metric projection onto convex closed sets. (English. Russian original) Zbl 0531.41029
Sov. Math., Dokl. 27, 330-332 (1983); translation from Dokl. Akad. Nauk SSSR 269, 289-291 (1983).
Let M be a nonempty set in a real normed linear space X. The multivalued mapping $$P_ M: x\to P_ Mx$$, defined by $$P_ Mx=\{y\in M: \| x- y\| =\inf\| x-z\|$$, $$z\in M\}$$, is called the metric projection of X onto M. The set M is a set of existence in X if $$P_ Mx\neq \emptyset$$. For such a set M in X various forms of continuity for the metric projection $$P_ M$$ can be defined (upper and lower semicontinuity, upper and lower H-semicontinuity, H-continuity and $$\rho$$-continuity). In a series of papers the present author [Mat. Sb., Nov. Ser. 80(122), 181-194 (1969; Zbl 0192.470), and Mat. Zametki 10, 459-468 (1971; Zbl 0234.46018)], N. V. Nevesenko [Mat. Zametki 23, 845-854 (1978; Zbl 0405.46011)] and both authors together [Dokl. Akad. Nauk SSSR 223, 1064-1066 (1975; Zbl 0341.46014) and Mat. Zametki 31, 117- 126 (1982)] obtained geometric characterizations for each of these forms of continuity for a metric projection.
In the present paper, the author points out ”that many of the geometric properties found were fairly complicated, and the relationships between them and the classical geometric properties of the unit sphere have remained obscure (in spite of) numerous attempts to clear them up.” In the present paper the author eludes these deficiencies and explaines some new relations between the various forms of continuity for a metric projection.
Reviewer: S.Aljančić

##### MSC:
 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)