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Semicontinuous functions and convex sets in C(K) spaces

Part of: Linear function spaces and their duals Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

J. P. Moreno
J. P. Moreno
Affiliation:
Dpto. Mateáticas Facultad de Ciencias Universidad Autónoma de MadridMadrid 28049Spain e-mail: [email protected]
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Abstract

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The stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that ℳ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C[0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 111 - 121
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Chakerian, G. D. and Groemer, H., ‘Convex bodies of constant width’, in: Convexity and its applications (eds. Gruber, P. and Wills, J.) (Birkhäuser, Basel, 1983) pp. 49–96.CrossRefGoogle Scholar
[2]Chen, D. and Lin, B.-L., ‘On B-convex and Mazur sets of Banach spaces’, Bull. Polish Acad. Sci. Math., 43 (1995), 191–198.Google Scholar
[3]Engelking, R., General topology, Sigma Series in Pure Mathematics 6 (Heldermann, Berlin, 1989).Google Scholar
[4]Frolik, Z., ‘Baire spaces and some generalizations of complete metric spaces’, Czechoslovak Math. J. 11 (1961), 237–248.CrossRefGoogle Scholar
[5]Granero, A. S., Moreno, J. P. and Phelps, R. R., ‘Convex sets which are intersection of closed balls’, Adv. Math. 183 (2004), 183–208.CrossRefGoogle Scholar
[6]Granero, A. S., Moreno, J. P. and Phelps, R. R., ‘Mazur sets in normed spaces’, Discrete Comput. Geom. 31 (2004), 411–420.CrossRefGoogle Scholar
[7]Sevilla, M. Jiménez and Moreno, J. P., ‘A note on porosity and the Mazur intersection property’, Mathematika 47 (2000), 267–272.CrossRefGoogle Scholar
[8]Sevilla, M. Jiménez and Moreno, J. P., ‘A constant of porosity for convex bodies’, Illinois J. Math. 45 (2001), 1061–1071.Google Scholar
[9]Meyer-Nieberg, P., Banach lattices (Springer, Berlin, 1991).CrossRefGoogle Scholar
[10]Moreno, J. P., Papini, P. L. and Phelps, R. R., ‘Diametrically maximal and constant width sets in Banach spaces’, Canad. J. Math. 58 (2006), 820–842.CrossRefGoogle Scholar
[11]Munkres, J. R., Topology: a first course (Prentice-Hall, Englewood Cliffs, N.J., 1975).Google Scholar
[12]Schneider, R., Convex bodies: The Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications 44 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar