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The two-ball property: transitivity and examples

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

Rafael Payá  and
David Yost
Rafael Payá
Affiliation:
Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.
David Yost
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, GPO Box 4 Canberra ACT 2601, Australia.
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Abstract

The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Mathematika , Volume 35 , Issue 2 , December 1988 , pp. 190 - 197
Copyright
Copyright © University College London 1988

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References

1.Alfsen, E. M. and Effros, E. G.. Structure in real Banach spaces. Ann. of Math., 96 (1972), 98–173.Google Scholar
2.Behrends, E.. M-structure and the Banach-Stone theorem. Lecture Notes in Math., 736 (Springer, 1979).Google Scholar
3.Cheney, E. W. and Wulbert, D. E.. The existence and unicity of best approximants. Math. Scand., 24 (1969), 113–140.CrossRefGoogle Scholar
4.Garkavi, A. L.. The conditional Chebyshev centre of a compact set of continuous functions (Russian). Mat. Zametki, 14 (1973), 469–478 ( = Math. Notes, 14 (1973), 827–831).Google Scholar
5.Godini, G.. Best approximation and intersections of balls. Lecture Notes in Math., 991 (Springer, 1983), 44–54.Google Scholar
6.Grünbaum, B.. On a conjecture of H. Hadwiger. Pacific J. Math., 11 (1961), 215–219.Google Scholar
7.Kelly, P. J.. On Minkowski bodies of constant width. Bull. Amer. Math. Soc., 55 (1949), 1147–1150.CrossRefGoogle Scholar
8.Klee, V.. What is a convex set? Amer. Math. Monthly, 78 (1971), 616–631.Google Scholar
9.Lima, A.. Intersection properties of balls in spaces of compact operators. Ann. Inst. Fourier, 28 (1978), 35–65.Google Scholar
10.Lindenstrauss, J.. Extension of compact operators. Mem. Amer. Math. Soc., 48 (1964).Google Scholar
11.Mena, J. F., Payá, R. and Rodríguez, A.. Semisummands and semiideals in Banach spaces. Israel J. Math., 51 (1985), 33–67.CrossRefGoogle Scholar
12.Mena, J. F., Payá, R., Rodríguez, A. and Yost, D. T.. Absolutely proximinal subspaces of Banach spaces. Preprint.Google Scholar
13.Shephard, G. C.. Decomposable convex polyhedra. Mathematika, 10 (1963), 89–95.CrossRefGoogle Scholar
14.Shephard, G. C.. Reducible convex sets. Mathematika, 13 (1966), 49–50.CrossRefGoogle Scholar
15.Soltan, V. P.. Bodies of constant width in finite dimensional normed spaces (Russian). Mat. Zametki, 25 (1979), 283–291 ( = Math. Notes, 25 (1979), 147–150).Google Scholar
16.Wong, Y. C. and Ng, K. F.. Partially ordered topological vector spaces (Clarendon Press, Oxford, 1973).Google Scholar
17.Yost, D.. The n-ball properties in real and complex Banach spaces. Math. Scand., 50 (1982), 100–110.CrossRefGoogle Scholar
18.Yost, D.. Banach spaces isomorphic to proper M-ideals. Colloq. Math., 56 (1988), 99–106.Google Scholar