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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

Information

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

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