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Points of symmetry of convex sets in the two-dimensional complex space – a counterexample to D. Yost’s problem. (English) Zbl 0757.46017
The 3-ball intersection property (\(M\)-ideal property) was known not to be equivalent to the 2-ball intersection property for real spaces. In the complex case the answer was not known. By a construction of D. Yost, the latter problem was known to be equivalent to the question whether only compact convex sets \(K\) in \(\mathbb{C}^ 2\) which have a center of symmetry have the property that \(f(K)\) is a disc for every linear map \(f:\mathbb{C}^ 2\to\mathbb{C}\).
The author constructs a nice counterexample: There is a set of the form \(K=\{(z,w)\mid\;| z|\leq 1\), \(| w|\leq\varphi(z)\}\) such that \(f(K)\) is a disc for every linear \(f:\mathbb{C}^ 2\to\mathbb{C}\), but has no center of symmetry. Consequently, the 3- and 2-ball properties differ in the complex case, too.
Reviewer: H.König (Kiel)

46B20 Geometry and structure of normed linear spaces
Full Text: DOI EuDML
[1] Alfsen, E.M., Effros, E.G.: Structure in real Banach spaces. Part I and II. Ann. Math.96, 98-173 (1972) · Zbl 0248.46019 · doi:10.2307/1970895
[2] Aupetit, B.: Analytic multivalued functions in Banach algebras and uniform algebras. Adv. Math.44, 18-60 (1982) · Zbl 0486.46041 · doi:10.1016/0001-8708(82)90064-0
[3] Aupetit, B.: Geometry of pseudoconvex open sets and distribution of values of analytic multivalued functions. Contemp. Math.32, 15-45 (1984) · Zbl 0595.32027
[4] Berndtsson, B., Ransford, T.J.: Analytic multifunctions the \(\bar \partial \) , and a proof of the Corona theorem. Pac. J. Math.124, 57-72 (1986) · Zbl 0602.32002
[5] Harmand, P., Werner, D., Werner, W.:M-ideals in Banach spaces and Banach algebras. 420 pages (Preprint) · Zbl 0789.46011
[6] Lima, ?.: Intersection properties of balls and subspaces in Banach spaces. Trans. Am. Math. Soc.227, 1-62 (1977) · Zbl 0347.46017 · doi:10.1090/S0002-9947-1977-0430747-4
[7] Lima, ?.: OnM-ideals and best approximation. Indiana Univ. Math. J.31, 27-36 (1982) · Zbl 0487.41042 · doi:10.1512/iumj.1982.31.31004
[8] Lima, ?., Yost, D.: Absolutely Chebyshev subspaces. In: Fitzpatrick, S., Giles, J. (eds.) Workshop/Miniconference Funct. Analysis/Optimization. Canberra 1988. Proc. Cent. Math. Anal. Aust. Natl. Univ.20, 116-127 (1988) · Zbl 0673.41035
[9] Mena-Jurado, J.F., Pay?, R., Rodr?guez-Palacios, A.: Absolute subspaces of Banach spaces. Quart. J. Math. Oxf.40, 43-64 (1989) · Zbl 0693.46014 · doi:10.1093/qmath/40.1.43
[10] Pay?, R., Rodr?guez-Palacios, A.: Banach spaces which are semi-L-summands in their biduals. Math. Ann.289, 529-542 (1991) · Zbl 0729.46008 · doi:10.1007/BF01446587
[11] Pay?, R., Yost, D.: The two-ball property: transitivity and examples. Matematika35, 190-197 (1988) · Zbl 0651.46024 · doi:10.1112/S0025579300015187
[12] Saatkamp, K.: Schnitteigenschaften und beste Approximation. Dissertation, Universit?t Bonn 1979
[13] Yost, D.: Then-ball properties in real and complex Banach spaces. Math. Scand.50, 100-110 (1982) · Zbl 0479.46008
[14] Yost, D.: Semi-M-ideals in complex Banach spaces. Rev. Roum. Math. Pures Appl.29, 619-623 (1984) · Zbl 0578.46015
[15] Yost, D.: Banach spaces isomorphic to properM-ideals. Colloq. Math.56, 99-106 (1988) · Zbl 0698.46013
[16] Yost, D.: Irreducible convex sets. Mathematika (1991) (to appear) · Zbl 0761.52011
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