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On a property of metric projections onto closed subsets of Hilbert spaces


Authors: J. Frerking and U. Westphal
Journal: Proc. Amer. Math. Soc. 105 (1989), 644-651
MSC: Primary 41A65; Secondary 41A52, 46C05, 47H05
DOI: https://doi.org/10.1090/S0002-9939-1989-0946636-6
MathSciNet review: 946636
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Abstract: Applying the theory of monotone operators to the metric projection $ {P_K}$ of a Hilbert space $ H$ onto a nonempty closed subset $ K$ of $ H$ we prove a kind of connectedness property of the set $ \{ x \in H;{P_K}(x)$ is not a singleton or $ {P_K}$ is not upper semi-continuous at $ x\} $ which is a typical set for investigations in best approximation. A result of Balaganskii is extended.


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  • [1] E. Asplund, Chebyshev sets in Hilbert space, Trans. Amer. Math. Soc. 144 (1969), 235-240. MR 0253023 (40:6238)
  • [2] V. S. Balaganskii, Approximative properties of sets in Hilbert space, Mat. Zametki 31 (1982), 785-800, Math. Notes 31 (1982), 397-404. MR 660584 (83m:41033)
  • [3] K. Bartke and H. Berens, Eine Beschreibung der Nichteindeutigkeitsmenge für die beste Approximation in der Euklidischen Ebene, J. Approx. Theory 47 (1986), 54-74. MR 843455 (87j:41069)
  • [4] H. Berens, Best approximation in Hilbert space, Approximation Theory III, edited by E. W. Cheney, Academic Press, New York, 1980, pp. 1-20. MR 602703 (82i:41030)
  • [5] -, Ein Problem über die beste Approximation in Hilberträumen, Functional Analysis and Approximation, edited by P. L. Butzer, B. Sz.-Nagy and E. Görlich, ISNM 60, Birkhäuser Basel, 1981, pp. 247-254. MR 650279 (83g:41036)
  • [6] H. Berens and U. Westphal, Kodissipative metrische Projektionen in normierten linearen Räumen, Linear Spaces and Approximation, edited by P. L. Butzer and B. Sz.-Nagy, ISNM 40, Birkhäuser Basel, 1978, pp. 120-130. MR 0511806 (58:23521)
  • [7] H. Brézis, Opérateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies no. 5, North-Holland Publishing Company, Amsterdam, 1973. MR 0348562 (50:1060)
  • [8] C. Franchetti and P. L. Papini, Approximation properties of sets with bounded complements, Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), 75-86. MR 628130 (82m:46016)
  • [9] G Godini, On a problem of H. Berens, Math. Ann. 263 (1983), 279-281. MR 704293 (84g:41031)
  • [10] P. S. Kenderov, A note on multivalued monotone mappings, C. R. Acad. Bulgare Sci. 28 (1975), 583-584. MR 0375007 (51:11203)
  • [11] V. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10-43. MR 0054850 (14:989d)
  • [12] -, Convexity of Chebyshev sets, Math. Ann. 142 (1961), 292-304.
  • [13] -, Dispersed Chebyshev sets and coverings by balls, Math. Ann. 257 (1981), 251-260. MR 634466 (84e:41036)
  • [14] Th. Motzkin, Sur quelques propriétés caractéristiques des ensembles convexes. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 21 (1935), 562-567.
  • [15] A. Pazy, Semi-groups of Nonlinear contractions in Hilbert space, Problems in Nonlinear Analysis, C.I.M.E. Varenna, Ed. Cremonese, Rome, 1971, pp. 345-430. MR 0291877 (45:965)
  • [16] S. B. Stečkin, Approximative properties of sets in linear normed spaces, Rev. Roumaine Math. Pures Appl. 8 (1963), 5-18. MR 0155168 (27:5108)
  • [17] L. P. Vlasov, On Chebyshev sets, Dokl. Akad. Nauk SSSR 173 (1967), 491-494 = Soviet Math. Dokl. 8 (1968), 401-409. MR 0215059 (35:5903)
  • [18] -, Approximative properties of sets in normed linear spaces, Uspekhi Mat. Nauk 28 (1973), 3-66 = Russian Math. Surveys 28 (1973), 1-66. MR 0404963 (53:8761)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0946636-6
Keywords: Best approximation in Hilbert spaces, convexity of Chebyshev sets, continuity of metric projections, uniqueness in best approximation, monotony of metric projections
Article copyright: © Copyright 1989 American Mathematical Society

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