Compacta with dense ambiguous loci of metric projections and antiprojections
Author:
N. V. Zhivkov
Journal:
Proc. Amer. Math. Soc. 123 (1995), 34033411
MSC:
Primary 41A65; Secondary 46B20, 54E52
MathSciNet review:
1273531
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In every strictly convexifiable Banach space X with there exists a dense set of compacta 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 are densely multivalued.
 [As]
Edgar
Asplund, Farthest points in reflexive locally uniformly rotund
Banach spaces, Israel J. Math. 4 (1966),
213–216. MR 0206662
(34 #6480)
 [Bl]
Jörg
Slatter, Weiteste Punkte und nächste Punkte, Rev.
Roumaine Math. Pures Appl. 14 (1969), 615–621
(German). 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), no. 12, 23–36. MR 1315681
(96d:52004), http://dx.doi.org/10.1007/BF01292766
 [BM1]
F.
S. De Blasi and J.
Myjak, Ambiguous loci of the nearest point mapping in Banach
spaces, Arch. Math. (Basel) 61 (1993), no. 4,
377–384. MR 1236316
(94i:41043), http://dx.doi.org/10.1007/BF01201454
 [BM2]
F.
S. De Blasi and J.
Myjak, Ambiguous loci of the farthest distance mapping from compact
convex sets, Studia Math. 112 (1995), no. 2,
99–107. MR
1311690 (95k:46020)
 [BM3]
, On compact connected sets in Banach spaces (to appear).
 [BF]
Jonathan
M. Borwein and Simon
Fitzpatrick, Existence of nearest points in Banach spaces,
Canad. J. Math. 41 (1989), no. 4, 702–720. MR 1012624
(90i:46024), http://dx.doi.org/10.4153/CJM19890327
 [Ko]
S.V. Konyagin, On approximation of closed sets in Banach spaces and the characterization of strongly convex spaces, Soviet Math. Dokl. 21 (1980), 418422.
 [L1]
Ka
Sing Lau, Farthest points in weakly compact sets, Israel J.
Math. 22 (1975), no. 2, 168–174. MR 0394126
(52 #14931)
 [L2]
Ka
Sing Lau, Almost Chebyshev subsets in reflexive Banach spaces,
Indiana Univ. Math. J. 27 (1978), no. 5,
791–795. MR 0510772
(58 #23286)
 [Lu]
D. Lubell, Proximity, Swiss cheese and offshore rights, preprint.
 [Si]
Ivan
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), 58.
 [Za]
Tudor
Zamfirescu, The nearest point mapping is single valued nearly
everywhere, Arch. Math. (Basel) 54 (1990),
no. 6, 563–566. MR 1052977
(91k:41061), http://dx.doi.org/10.1007/BF01188685
 [Zh]
N.
V. Zhivkov, Examples of plane compacta with dense ambiguous
loci, C. R. Acad. Bulgare Sci. 46 (1993), no. 1,
27–30. MR
1264020 (95e:52007)
 [Zh2]
N.
V. Zhivkov, Peano continua generating densely multivalued metric
projections, Rend. Sem. Mat. Univ. Politec. Torino 52
(1994), no. 4, 335–346. MR 1345603
(97a:46017)
 [As]
 E. Asplund, Farthest points in reflexive locally uniformly rotund spaces, Israel J. Math. 4 (1966), 213216. MR 0206662 (34:6480)
 [Bl]
 J. Blatter, Weiteste Punkte und Nachste Punkte, Rev. Roumaine Math. Pures Appl. 14 (1969), 615621. 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), 2336. 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), 377384. 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), 702720. 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), 418422.
 [L1]
 KaSing Lau, Farthest points in weakly compact spaces, Israel J. Math. 22 (1975), 168174. MR 0394126 (52:14931)
 [L2]
 , Almost Chebyshev subsets in reflexive Banach spaces, Indiana Univ. Math. J. 27 (1978), 791795. 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), 167177. MR 0178450 (31:2707)
 [St]
 S.B. Stechkin, Approximative properties of subsets of Banach spaces, Rev. Roumaine Math. Pures Appl. 8 (1963), 58.
 [Za]
 T. Zamfirescu, The nearest point mapping is singlevalued nearly everywhere, Arch. Math. 54 (1990), 563566. MR 1052977 (91k:41061)
 [Zh]
 N.V. Zhivkov, Examples of plane compacta with dense ambiguous loci, C. R. Acad. Bulgare Sci. 46 (1993), 2730. MR 1264020 (95e:52007)
 [Zh2]
 , Peano continua generating densely multivalued metric projections, Rend. Sem. Mat. Univ. Politec. Torino (to appear). MR 1345603 (97a:46017)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
41A65,
46B20,
54E52
Retrieve articles in all journals
with MSC:
41A65,
46B20,
54E52
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512735310
PII:
S 00029939(1995)12735310
Keywords:
dense ,
metric projection,
antiprojection,
ambiguous locus
Article copyright:
© Copyright 1995 American Mathematical Society
