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Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions

Published online by Cambridge University Press:  20 November 2018

Mark A. Smith
Mark A. Smith*
Affiliation:
Lake Forest College, Lake Forest, Illinois 60045; Miami University, Oxford, Ohio, 45056
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In a Banach space, the directional modulus of rotundity, δ (ϵ, z), measures the minimum depth at which the midpoints of all chords of the unit ball which are parallel to z and of length at least ϵ are buried beneath the surface. A Banach space is uniformly rotund in every direction (URED) if δ (ϵ, z) is positive for every positive ϵ and every nonzero element z. This concept of directionalized uniform rotundity was introduced by Garkavi [6] to characterize those Banach spaces in which every bounded subset has at most one Čebyšev center.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 5 , 01 October 1977 , pp. 963 - 970
Copyright
Copyright © Canadian Mathematical Society 1977

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