## Annals of Functional Analysis

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

#### 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 $P_{C}(x)\neq\emptyset$, 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
Accepted: 17 May 2016
First available in Project Euclid: 14 October 2016

https://projecteuclid.org/euclid.afa/1476450344

Digital Object Identifier
doi:10.1215/20088752-3720520

Mathematical Reviews number (MathSciNet)
MR3558301

Zentralblatt MATH identifier
1368.46018

#### 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

#### References

• [1] P. Bandyopadhyay, D. Huang, and B.-L. Lin, Rotund points, nested sequence of ball and smoothness in Banach space, Comment Math. 44 (2004), no. 2, 163–186.
• [2] P. Bandyopadhyay, D. Huang, B.-L. Lin, and S. T. Troyanski, Some generalizations of locally uniform rotundity, J. Math. Anal. Appl. 252 (2000), no. 2, 906–916.
• [3] P. Bandyopadhyay, Y. Li, B.-L. Lin, and D. Naraguna, Proximinality in Banach spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 309–317.
• [4] X. Fang and J. Wang, Convexity and continuity of metric projection, Math. Appl. 14 (2001), no. 1, 47–51.
• [5] J. R. Giles, P. S. Kenderow, W. B. Moors, and S. D. Sciffer, Generic differentiability of convex functions on the dual of a Banach space, Pacific J. Math. 172 (1996), no. 2, 413–431.
• [6] A. J. Guirao and V. Montesinos, A note in approximative compactness and continuity of metric projections in Banach spaces, J. Convex Anal. 18 (2011), no. 2, 397–401.
• [7] Z. Hu and B.-L. Lin, Smoothness and the asymptotic-norming properties of Banach spaces, Bull. Austral. Math. Soc. 45 (1992), no. 2, 285–296.
• [8] C. Nan and J. Wang, On the Lk-UR and L-kR spaces, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 3, 521–526.
• [9] B. B. Panda and O. P. Kapoor, A generalization of local uniform convexity of the norm, J. Math. Anal. Appl. 52 (1975), 300–305.
• [10] J. Wang and C. Nan, On the dual spaces of the S-spaces and WkR spaces, Chinese J. Contemp. Math. 13 (1992), no. 1, 23–27.
• [11] J. Wang and Z. Zhang, Characterizations of the property $(C-\kappa)$, Acta Math. Sci. Ser. A Chin. Ed. 17 (1997), no. 3, 280–284.
• [12] Z. Zhang and Z. Shi,Convexities and approximative compactness and continunity of metric projection in Banach spaces, J. Approx. Theory. 161 (2009), no. 2, 802–812.