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Normed Linear Spaces that are Uniformly Convex in Every Direction

Published online by Cambridge University Press:  20 November 2018

M. M. Day ,
R. C. James  and
S. Swaminathan
M. M. Day
Affiliation:
University of Illinois, Urbana, Illinois
R. C. James
Affiliation:
Claremont Graduate School, Claremont, California
S. Swaminathan
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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The concept of uniform convexity in a normed linear space is based on the geometric condition that if two members of the unit ball are far apart, then their midpoint is well inside the unit ball. We consider here a generalization of this concept whose geometric significance is that the collection of all chords of the unit ball that are parallel to a fixed direction and whose lengths are bounded below by a positive number has the property that the midpoints of the chords lie uniformly deep inside the unit ball. This notion, called uniform convexity in every direction (UCED), was first used by A. L. Garkavi [5; 6] to characterize normed linear spaces for which every bounded subset has at most one Čebyŝev center. We discuss questions of renorming spaces so as to be UCED and forming products of spaces that are uniformly convex in every direction.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 6 , December 1971 , pp. 1051 - 1059
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Belluce, L. P. and Kirk, W. A., Nonexpansive mappings and fixed-point s in Banach spaces, Illinois J. Math. 11 (1967), 474–479.Google Scholar
2. Belluce, L. P., Kirk, W. A., and Steiner, E. F., Normal structure in Banach spaces, Pacific J. Math. 26 (1968), 433–440.Google Scholar
3. Brodskii, M. S. and Milman, D. P., On the center of a convex set, Dokl. Akad. Nauk SSSR (N.S.) 59 (1948), 837–840.Google Scholar
4. Day, M. M., Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528.Google Scholar
5. Garkavi, A. L., On the Cebysev center of a set in a normed space, Investigations of Contemporary Problems in the Constructive Theory of Functions, Moscow, 1961, pp. 328–331.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. James, R. C., Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550.Google Scholar
8. James, R. C., Super-reflexive spaces with bases (to appear in Pacific J. Math.).Google Scholar
9. Zizler, V., Some notes on various rotundity and smoothness properties of separable Banach spaces, Comment, Math. Univ. Carolinae 10 (1969), 195–206.Google Scholar
10. Zizler, V., On some rotundity and smoothness properties of Banach spaces (to appear in Dissertationes Math. Rozprawy Mat.).Google Scholar