On quasi-denting points, denting faces and the geometry of the unit ball of \(d(w,1)\)
type: Item
instance of: scholarly article
title: On quasi-denting points, denting faces and the geometry of the unit ball of \(d(w,1)\)
zbMATH Open document ID: 0805.46020
DOI: 10.1007/BF01196298
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: Some methods to characterize the quasi-denting points of a closed convex bounded set \(C\) in a Banach space are discussed. This property of points has been recently introduced by R. Giles and W. B. Moors, and its relevance increased by the fact (proved by the third author of the paper) that a Banach space such that every point of the unit sphere is quasi- denting can be renormed to be LUR. It is known that the denting points of \(C\) are its extreme points of continuity, nevertheless in the paper it is shown that a similar statement does not hold for quasi-denting points. However the characterization of denting points through Bochner integrable functions is extended to quasi-denting points. The concept of denting face introduced in this paper, provides another approach to check this property, since quasi-denting points of such faces are quasi-denting for the whole set \(C\). The authors show that every face with the Krein-Milman property, made up by points of continuity, is denting. From these results they deduce that all faces of the unit sphere \(S\) of the Lorentz sequence spaces \(d(w,1)\) are denting, and all the points of \(S\) are quasi-denting for the unit ball of \(d(w,1)\).
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 687085
zbMATH Keywords: quasi-denting points of a closed convex bounded set
zbMATH Keywords: LUR
zbMATH Keywords: denting face
zbMATH Keywords: Krein-Milman property
author: A. Moltó
author: V. Montesinos
author: S. Troyanski
MaRDI profile type: Publication
cites work: Uniform convexity in Lorentz sequence spaces
cites work: Q3954333
cites work: Some geometric properties of Lorentz sequence spaces
cites work: Q4283259
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cites work: On the structure of non-dentable closed bounded convex sets
cites work: On some generalizations of denting points
full work available at URL: https://doi.org/10.1007/bf01196298
OpenAlex ID: W1994761550
description: scientific article; zbMATH DE number 687085