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Compacta with dense ambiguous loci of metric projections and antiprojections

Author: N. V. Zhivkov
Journal: Proc. Amer. Math. Soc. 123 (1995), 3403-3411
MSC: Primary 41A65; Secondary 46B20, 54E52
MathSciNet review: 1273531
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Abstract: In every strictly convexifiable Banach space X with $ \dim X \geq 2$ there exists a dense $ {G_\delta }$ set of compacta $ \mathcal{A}$ in the Hausdorff set topology such that with respect to an arbitrary equivalent strictly convex norm in X both the metric projection and the metric antiprojection generated by any member of $ \mathcal{A}$ are densely multivalued.

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  • [As] E. Asplund, Farthest points in reflexive locally uniformly rotund spaces, Israel J. Math. 4 (1966), 213-216. MR 0206662 (34:6480)
  • [Bl] J. Blatter, Weiteste Punkte und Nachste Punkte, Rev. Roumaine Math. Pures Appl. 14 (1969), 615-621. MR 0251510 (40:4737)
  • [BKM] F.S. De Blasi, P.S. Kenderov, and J. Myjak, Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space, Monatsh. Math. 119 (1995), 23-36. MR 1315681 (96d:52004)
  • [BM1] F.S. De Blasi and J. Myjak, Ambiguous loci of the nearest point mapping in Banach spaces, Arch. Math. 61 (1993), 377-384. MR 1236316 (94i:41043)
  • [BM2] -, Ambiguous loci of the farthest distance mapping from compact convex sets, Studia Math. (to appear). MR 1311690 (95k:46020)
  • [BM3] -, On compact connected sets in Banach spaces (to appear).
  • [BF] J.M. Borwein and S. Fitzpatrick, Existence of nearest points in Banach spaces, Canad. J. Math. 41 (1989), 702-720. MR 1012624 (90i:46024)
  • [Ko] S.V. Konyagin, On approximation of closed sets in Banach spaces and the characterization of strongly convex spaces, Soviet Math. Dokl. 21 (1980), 418-422.
  • [L1] Ka-Sing Lau, Farthest points in weakly compact spaces, Israel J. Math. 22 (1975), 168-174. MR 0394126 (52:14931)
  • [L2] -, Almost Chebyshev subsets in reflexive Banach spaces, Indiana Univ. Math. J. 27 (1978), 791-795. MR 0510772 (58:23286)
  • [Lu] D. Lubell, Proximity, Swiss cheese and offshore rights, preprint.
  • [Si] I. Singer, Some remarks on approximative compactness, Rev. Roumaine Math. Pures Appl. 9 (1964), 167-177. MR 0178450 (31:2707)
  • [St] S.B. Stechkin, Approximative properties of subsets of Banach spaces, Rev. Roumaine Math. Pures Appl. 8 (1963), 5-8.
  • [Za] T. Zamfirescu, The nearest point mapping is single-valued nearly everywhere, Arch. Math. 54 (1990), 563-566. MR 1052977 (91k:41061)
  • [Zh] N.V. Zhivkov, Examples of plane compacta with dense ambiguous loci, C. R. Acad. Bulgare Sci. 46 (1993), 27-30. MR 1264020 (95e:52007)
  • [Zh2] -, Peano continua generating densely multivalued metric projections, Rend. Sem. Mat. Univ. Politec. Torino (to appear). MR 1345603 (97a:46017)

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Keywords: dense $ {G_\delta }$, metric projection, antiprojection, ambiguous locus
Article copyright: © Copyright 1995 American Mathematical Society

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