Complutense University Library

Antiproximinal norms in Banach spaces

Borwein , Jonathan M. and Jiménez Sevilla, María del Mar and Moreno, José Pedro (2002) Antiproximinal norms in Banach spaces. Journal of Approximation Theory, 114 (1). pp. 57-69. ISSN 0021-9045

[img] PDF
Restricted to Repository staff only until 31 December 2020.

135kB

Official URL: http://www.sciencedirect.com/science/article/pii/S0021904501936366

View download statistics for this eprint

==>>> Export to other formats

Abstract

We prove that every Banach space containing a complemented copy of c0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal norms in Banach spaces with the Convex Point of Continuity Property (CPCP). Other questions related to the existence of antiproximinal bodies are also discussed.

Item Type:Article
Additional Information:This work was begun while the second and third authors were visiting the CECM at the Simon Fraser University. The second author is indebted to Gilles Godefroy for his support, valuable suggestions, and many stimulating conversations.
Uncontrolled Keywords:Convex-functions; Intersection-properties; Continuity property; Differentiability; Sets; Point
Subjects:Sciences > Mathematics > Numerical analysis
ID Code:16414
References:

V.S. Balaganskii. Antiproximinal sets in spaces of continuous functions. Math. Notes, 60 (1996), pp. 485–494

R.D. Bourgin. Geometric Aspects of Convex Sets with the Radon–Nikodým Property, Lecture Notes in Math., 993Amer. Math. Soc, Providence (1983)

Dongjian Chen, Zhibao Hu, Bor-Luh Lin. Ball intersection properties of Banach spaces. Bull. Austral. Math. Soc., 45 (1992), pp. 333–342

S. Cobzas. Convex antiproximinal sets in the spaces c0 and c.Mat. Zametki, 17 (1975), pp. 449–457

S. Cobzas. Antiproximinal sets in Banach spaces of continuous functions. Rev. Anal. Numér. Théor. Approx., 5 (1976), pp. 127–143

S. Cobzas. Antiproximinal sets in Banach spaces of c0-type. Rev. Anal. Numér. Théor. Approx., 7 (1978), pp. 141–145

S.Cobzas.Antiproximinal sets in the Banach space c(X). Comment. Math. Univ. Carolin., 38 (1997), pp. 247–253

S. Cobzas. Multimifoarte neproximinale in c0. Rev. Anal. Numér. Théor. Approx., 2 (1973), pp. 137–141

R. Deville, G. Godefroy, Hare, V. Zizler. Differentiability of convex functions and the convex point of continuity property in Banach spaces. Israel J. Math., 59 (1987), pp. 245–255

R. Deville, P. Hajek, V. Fonf. Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces. Israel J. Math., 105 (1998), pp. 139–154

M. Edelstein. Antiproximinal sets. J. Approx. Theor., 49 (1987), pp. 252–255

M. Edelstein, A.C. Thompson. Some results on nearest point and support properties of convex sets in c0. Pacific J. Math., 40 (1972), pp. 553–560

V.P. Fonf. Some properties of polyhedral Banach spaces. Funktsional. Anal. i Prilozhen., 14 (1980), pp. 89–90

V.P. Fonf. On antiproximinal sets in spaces of continuous functions on compacta. Mat. Zametki, 33 (1983), pp. 549–558

V.P.Fonf.Strongly antiproximinal sets in Banach spaces.Mat. Zametki,47 (1990), pp. 130–136

P.G. Georgiev.Fréchet differentiability of convex functions in separable Banach spaces. C. R. Acad. Bulgare Sci., 43 (1990), pp. 13–15

P.G.Georgiev,A.S. Granero, M. Jiménez-Sevilla, J.P. Moreno. Mazur intersection property and differentibility of convex functions. J. London Math. Soc. (2), 61 (2000), pp. 531–542

N. Ghoussoub, B. Maurey, W. Schachermayer. A counterexample to a problem on points of continuity in Banach spaces. Proc. Amer. Math. Soc., 99 (1987), pp. 278–282

M. Jiménez-Sevilla, J.P. Moreno. A note on norm attaining functionals. Proc. Amer. Math. Soc.,126(1998), pp. 1989–1997

V. Klee, Remarks on nearest points in normed linear spaces, in Proc.Colloquium on Convexity,Copenhagen,1965,pp. 168–176.

J. Lindenstrauss, L. Tzafriri. Classical Banach Spaces. I. Sequences SpacesSpringer-Verlag, Berlin (1979)

W.B. Moors. The relationship between Goldstine's Theorem and the convex point of continuity property. J. Math. Anal. Appl., 188 (1994), pp. 819–832

J.P. Moreno. Geometry of Banach spaces with property α. J. Math. Anal. Appl., 201 (1996), pp. 600–608

R.R. Phelps. Counterexamples concerning support theorems for convex sets in Hilbert spaces. Canad. Math. Bull., 31 (1988), pp. 121–128

H.H.Shaefer.Banach Lattices and Positive. OperatorsSpringer-Verlag, New York/Berlin (1974)

J. Vanderwerff. Fréchet differentiable norms on spaces of countable dimension. Arch. Math., 58 (1992), pp. 471–476

Deposited On:18 Sep 2012 08:39
Last Modified:07 Feb 2014 09:29

Repository Staff Only: item control page