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Averaging distances in certain Banach spaces

Published online by Cambridge University Press:  17 April 2009

Reinhard Wolf
Reinhard Wolf
Affiliation:
Institut für Mathematik, Universität Salzburg, Hellbrunnerstraße 34, A-5020 Salzburg, Austria
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Abstract

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Let E be a Banach space. The averaging interval AI(E) is defined as the set of positive real numbers α, with the following property: For each n ∈ ℕ and for all (not necessarily distinct) x1, x2, … xn ∈ E with ∥x1∥ = ∥x2∥ = … = ∥xn∥ = 1, there is an x ∈ E, ∥x∥ = 1, such that

It follows immediately, that AI(E) is a (perhaps empty) interval included in the closed interval [1,2]. For example in this paper it is shown that AI(E) = {α} for some 1 < α < 2, if E has finite dimension. Furthermore a complete discussion of AI(C(X)) is given, where C(X) denotes the Banach space of real valued continuous functions on a compact Hausdorff space X. Also a Banach space E is found, such that AI(E) = [1,2].

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 1 , February 1997 , pp. 147 - 160
Copyright
Copyright © Australian Mathematical Society 1997

References

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