Annals of Functional Analysis

Characterizations and applications of three types of nearly convex points

Zihou Zhang, Yu Zhou, and Chunyan Liu

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Abstract

By using some geometric properties and nested sequence of balls, we prove seven necessary and sufficient conditions such that a point x in the unit sphere of Banach space X is a nearly rotund point of the unit ball of the bidual space. For any closed convex set CX and xXC with PC(x), we give a series of characterizations such that C is approximatively compact or approximatively weakly compact for x by using three types of nearly convex points. Furthermore, making use of an S point, we present a characterization such that the convex subset C is approximatively compact for some x in XC. We also establish a relationship between nested sequence of balls and the approximate compactness of the closed convex subset C for some xXC.

Article information

Source
Ann. Funct. Anal., Volume 8, Number 1 (2017), 16-26.

Dates
Received: 14 January 2016
Accepted: 17 May 2016
First available in Project Euclid: 14 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1476450344

Digital Object Identifier
doi:10.1215/20088752-3720520

Mathematical Reviews number (MathSciNet)
MR3558301

Zentralblatt MATH identifier
1368.46018

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Keywords
nearly rotund point nearly very convex point S point approximatively weak compactness nested sequence of balls

Citation

Zhang, Zihou; Zhou, Yu; Liu, Chunyan. Characterizations and applications of three types of nearly convex points. Ann. Funct. Anal. 8 (2017), no. 1, 16--26. doi:10.1215/20088752-3720520. https://projecteuclid.org/euclid.afa/1476450344


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