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Continuity of the metric projection on convex sets. (English. Russian original) Zbl 0816.46011
Math. Notes 52, No. 6, 1173-1177 (1992); translation from Mat. Zametki 52, No. 6, 3-9 (1992).
Summary: The continuity of the metric projection on convex sets has been studied since the fifties and sixties. A survey of these investigations has been given by E. V. Oshman [Mat. Zametki 37, No. 2, 200-211 (1985; Zbl 0586.41026)]. In this note we present a series of additional considerations, and we refine some isolated results.
MSC:
46B20 Geometry and structure of normed linear spaces
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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[1] E. V. Oshman, ?On the continuity of the metric projection,? Mat. Zametki,37, No. 2, 200-211 (1985). · Zbl 0586.41026
[2] L. P. Vlasov, ?The concept of approximative compactness and its variants,? Mat. Zametki,16, No. 2, 337-348 (1974).
[3] E. V. Oshman, ?On the continuity of metric projection in a Banach space,? Mat. Sb.,80 (122), 181-194 (1969).
[4] E. V. Oshman, ?On the continuity of the metric projection onto convex closed sets,? Dokl. Akad. Nauk SSSR,269, No. 2, 289-291 (1983). · Zbl 0531.41029
[5] E. V. Oshman, ?Chebyshev sets, continuity of the metric projection and certain geometric properties of the unit sphere in a Banach space,? Izv. Vyssh. Uchebn. Zaved., Matematika, No. 4, 38-46 (1969).
[6] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton (1970). · Zbl 0193.18401
[7] E. Bishop and R. R. Phelps, ?A proof that every Banach space is subreflexive,? Bull. Am. Math. Soc,67, No. 1, 97-98 (1961). · Zbl 0098.07905 · doi:10.1090/S0002-9904-1961-10514-4
[8] L. P. Vlasov, ?Approximative properties of sets in normed linear spaces,? Uspekhi Mat. Nauk,28, No. 6, 3-66 (1973). · Zbl 0293.41031
[9] N. V. Nevesenko, ??-continuity of the metric projection onto convex closed sets,? Mat. Zametki,23, No. 6, 845-854 (1978).
[10] N. V. Nevesenko and E. V. Oshman, ?Metric projection into convex sets,? Mat. Zametki,31, No. 1, 117-126 (1982). · Zbl 0578.41038
[11] L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977). · Zbl 0127.06102
[12] L. P. Vlasov, Properties of generalized elements of the best approximation,? Mat. Zametki,24, No. 4, 513-522 (1978). · Zbl 0419.41026
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