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Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

Published online by Cambridge University Press:  20 November 2018

Mark A. Smith
Mark A. Smith*
Affiliation:
Lake Forest College, Lake Forest, Illinois 60045; Miami University, Oxford, Ohio, 45056
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In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 5 , 01 October 1977 , pp. 963 - 970
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Clarkson, J. A., Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.Google Scholar
2. Day, M. M., Normed linear spaces, 3rd éd., Ergebnisse der Mathematik und ihrer Grenzgebiete, 21 (Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
3. Day, M. M. Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528.Google Scholar
4. Day, M. M., James, R. C., and Swaminathan, S., Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23 (1971), 1051–1059.Google Scholar
5. Fakhoury, H., Directions d'uniforme convexité dans un espace norme, Séminaire Choquet (14e année: 1974/75). Initiation à l'analyse, Exp. No. 6, 16pp (Secrétariat Mathématique, Paris, 1975).Google Scholar
6. Garkavi, A. L., The best possible net and the best possible cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87-106; Amer. Math. Soc. Transi., Ser. 2, 39 (1964), 111–132.Google Scholar
7. Klee, V. L., Jr., Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51–63.Google Scholar
8. Smul'yan, V. L., Sur la dêrivabilité de la norme dans Vespace de Banach, C. R. (Doklady) Acad. Sci. URSS 27 (1940), 643–648.Google Scholar
9. Troyanski, S. L., On non-separable Banach spaces with a symmetric basis, Studia Math. 53 (1975), 253–263.Google Scholar
10. Zizler, V., On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) No. 87 (1971).Google Scholar