Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T05:46:06.639Z Has data issue: false hasContentIssue false
  • English
  • Français

On the geometry of spheres in normed linear spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  09 April 2009

Ji Gao  and
Ka-Sing Lau
Ji Gao
Affiliation:
University of Pittsburgh Pittsburgh, Pennsylvania, U.S.A.
Ka-Sing Lau
Affiliation:
University of Pittsburgh Pittsburgh, Pennsylvania, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Some simplifications of Schäffer's girth and perimeter of the unit spheres are introduced. Their general properties are discussed, and they are used to study the lp, Lp spaces, uniformly nonsquare spaces, and their isomorphic classes.

Keywords

girthisomorphismperimeteruniformly convexuniformly nonsquare.

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 1 , February 1990 , pp. 101 - 112
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Clarkson, J. A., ‘Uniformly convex spaces’, Trans. Amer. Math. Soc. 40 (1936), 396–414.CrossRefGoogle Scholar
[2]Day, M. M., Normed liner spaces (Springer-Verlag, New York-Heidelberg-Berlin, 1973, 2nd ed.).CrossRefGoogle Scholar
[3]Gao, J., ‘The uniform degree of the unit ball of a Banach space’, Nanjing Daxue Xuebao I (1982), 14–28 (in Chinese).Google Scholar
[4]Gao, J., ‘The extreme value of the uniform degree of the unit ball in a Banach space’, Nanjing Daxue Xuebao I (1983), 5–12 (in Chinese).Google Scholar
[5]James, R. C., ‘Uniformly non-square spaces’, Ann. of Math. 80 (1964), 542–550.CrossRefGoogle Scholar
[6]James, R. C. and Schäffer, J. J., ‘Superreflexivity and the girth of spheres’, Israel J. Math. 11 (1972), 398–402.CrossRefGoogle Scholar
[7]Nyikos, P. and Schäffer, J. J., ‘Flat spaces of continuous functions’, Studia Math. 42 (1972), 221–229.CrossRefGoogle Scholar
[8]Schäffer, J. J., ‘On the geometry of spheres in L spaces’, Israel J. Math. 10 (1971), 114–120.CrossRefGoogle Scholar
[9]Schäffer, J. J., ‘On the geometry of spheres of continuous functions,’ J. Analyse Math. 26 (1973), 337–389.CrossRefGoogle Scholar
[10]Schäffer, J. J., Geometry of spheres in normed spaces (Marcel Dekker, New York, 1976).Google Scholar
[11]Schäffer, J. J. and Sundaresen, , ‘Reflexivity and the girth of spheres, math. Ann. 184 (1970), 163–168.CrossRefGoogle Scholar
[12]Borwein, J. and Keener, L., ‘The Hausdorff metric and Čebyšev centres’, J. Approx. Theory 28 (1980), 366–376.CrossRefGoogle Scholar
[13]Gao, J. and Lau, K. S., ‘On two classes of Banach spaces with normal structure’, submitted.Google Scholar