×

zbMATH — the first resource for mathematics

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)

MSC:
46B20 Geometry and structure of normed linear spaces
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.