## Annals of Functional Analysis

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

### Characterizations and applications of three types of nearly convex points

Zihou Zhang, Yu Zhou, and Chunyan Liu

#### 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 $C\subset X$ and $x\in X\setminus C$ with ${P}_{C}\left(x\right)\ne \varnothing $, 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 $X\setminus C$. We also establish a relationship between nested sequence of balls and the approximate compactness of the closed convex subset $C$ for some $x\in X\setminus C$.

#### 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