Asymptotic geometry and delta-points

We study Daugavet- and \(\Delta\)-points in Banach spaces. A norm one element \(x\) is a Daugavet-point (respectively a \(\Delta\)-point) if in every slice of the unit ball (respectively in every slice of the unit ball containing \(x\)) you can find another element of distance as close to \(2\) from...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Abrahamsen, Trond A, Lima, Vegard, Martiny, André, Perreau, Yoël
Format: Article
Language:English
Subjects:
Asymptotic properties
Banach spaces
Online Access:Get full text
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Summary:We study Daugavet- and \(\Delta\)-points in Banach spaces. A norm one element \(x\) is a Daugavet-point (respectively a \(\Delta\)-point) if in every slice of the unit ball (respectively in every slice of the unit ball containing \(x\)) you can find another element of distance as close to \(2\) from \(x\) as desired. In this paper we look for criteria and properties ensuring that a norm one element is not a Daugavet- or \(\Delta\)-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain \(\Delta\)-points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally we prove that there exists a superreflexive Banach space with a Daugavet- or \(\Delta\)-point provided there exists such a space satisfying a weaker condition.
ISSN:2331-8422