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Flat spots on unit spheres

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

Published online by Cambridge University Press:  09 April 2009

D. Van Dulst
D. Van Dulst
Affiliation:
Mathematisch Instituut Universiteit van AmsterdamRoetersstraat 15, Amsterdam, Holland
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Abstract

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A flat spot in a Banach space X is an element x ∈ Sx = {x ∈ X: ‖x‖ = 1} with the property that the infimum m(x) of the lengths of all curves in Sx joining x to −x is 2. Flat spots occur in every non-superreflexive space when suitably renormed. A study is made of the geometric implications of the existence of flat spots. Connections with other notions such as differentiability, decomposition constants and Kadec-Klee norms are explored and some renorming results for non-superreflexive spaces are proved.

MSC classification

Secondary: 46B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 3 , May 1979 , pp. 289 - 304
Copyright
Copyright © Australian Mathematical Society 1979

References

Amir, D. and Lindenstrauss, J. (1968), “The structure of weakly compact sets in Banach spaces”, Ann. of Math. 88, 35–46.Google Scholar
Asplund, E. (1968), “Fréchet differentiability of convex functions”, Acta Math. 221, 31–48.CrossRefGoogle Scholar
Bishop, E. and Phelps, R. R. (1961), “A proof that every Banach space is subreflexive”, Bull. Amer. Math. Soc. 67, 97–98.Google Scholar
Van Dulst, D. and Schäffer, J. J. (1979), “Two examples of flat spots in non-flat Banach spaces”, Nieuw Arch. Wisk. (to appear).Google Scholar
Enflo, P. (1972), “Banach spaces which can be given an equivalent uniformly convex norm”, Israel J. Math. 13, 281–288.Google Scholar
Figiel, T. and Johnson, W. B. (1974), “A uniformly convex Banach space which contains no lv”, Comp. Math. 29, 179–190.Google Scholar
Harrell, R. E. and Karlovitz, L. A. (1972), “Non-reflexivity and the girth of spheres, Inequalities III”, Acad. Press, 121–127.Google Scholar
Harrell, R. E. and Karlovitz, L. A. (1974), “The geometry of flat Banach spaces”, Trans. Amer. Math. Soc. 192, 209–218.Google Scholar
James, R. C. (1972), “Some self-dual properties of normed linear spaces. Symposium on infinite dimensional topology”, Ann. of Math. Studies 69, 159–175.Google Scholar
James, R. C. and Schäffer, J. J. (1972), “Superreflexivity and the girth of spheres”, Israel J. Math. 11, 398–404.CrossRefGoogle Scholar
John, K. and Zizler, V. (1972), “A renorming of dual spaces”, Israel J. Math. 12, 331–336.CrossRefGoogle Scholar
John, K. and Zizler, V. (1974), “Smoothness and its equivalents in weakly compactly generated Banach spaces”, J. Functional Analysis 15, 1–11.CrossRefGoogle Scholar
Johnson, W. B., Rosenthal, H. P. and Zippin, M. (1971), “On bases, finite-dimensional decompositions and weaker structures in Banach spaces”, Israel J. Math. 9, 488–506.CrossRefGoogle Scholar
Kadec, M. I. (1952), “On the connection between weak and strong convergence”, Dopovidi Akad. Nauk Ukrafn. RSR 9, 949–952 (Ukrainian).Google Scholar
Karlovitz, L. A. (1973), “On the duals of flat Banach spaces”, Math. Ann. 202, 245–250.CrossRefGoogle Scholar
Karlovitz, L. A. (1976), “On non-expansive mappings”, Proc. Amer. Math. Soc. 55, 321–325.CrossRefGoogle Scholar
Klee, V. (1960/1961), “Mappings into normed linear spaces”, Fund. Math. 49, 25–34.CrossRefGoogle Scholar
Lindenstrauss, J. (1972), “Weakly compact sets—their topological properties and the Banach spaces they generate”, Ann. of Math. Studies 69, 235–273.Google Scholar
Lindenstrauss, J. and Rosenthal, H. P. (1969), “The ℒv spaces”, Israel J. Math. 7, 325–349.CrossRefGoogle Scholar
Namioka, I. and Phelps, R. R. (1975), “Banach spaces which are Asplund spaces”, Duke Math. J. 42, 735–750.Google Scholar
Pach, A. J. (1979), “Flat spots and super-reflexivity in normed spaces (preprint, University of Amsterdam).Google Scholar
Rainwater, J. (1969), “Local uniform convexity of Day's norm on c0(Ѓ)”, Proc. Amer. Math. Soc. 22, 335–339.Google Scholar
Schäffer, J. J. (1967), “Inner diameter, perimeter and girth of spheres”, Math. Ann. 173, 59–79.Google Scholar
Schäffer, J. J. (1971), “On the geometry of spheres in L-spaces”, Israel J. Math. 10, 114–120.CrossRefGoogle Scholar
Schäffer, J. J. (1976), Geometry of spheres in normed spaces (Lecture notes in pure and applied mathematics, Marcel Dekker, New York and Basel).Google Scholar
Stegall, C. (1977), “The Radon-Nikodym property in conjugate spaces II” (preprint).Google Scholar
Troyanski, S. (1971), “On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces”, Studia Math. 37, 173–180.CrossRefGoogle Scholar
Tsirelson, B. S. (1974), “Not every Banach space containslp or c0”, Functional Anal. Appl. 8, 138–141 (translated from Russian).CrossRefGoogle Scholar