zbMATH — the first resource for mathematics

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.
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)
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.