Journal of Convex Analysis 25 (2018), No. 3, 1045--1058
Copyright Heldermann Verlag 2018
Dentable Point and Ball-Covering Property in Banach Spaces
Shaoqiang Shang
Dept. of Mathematics, Northeast Forestry University, Harbin 150040, P. R. China
sqshang@163.com
Yunan Cui
Dept. of Mathematics, Harbin University of Science and Technology, Harbin 150080, P. R. China
cuiya@hrbust.edu.cn
[Abstract-pdf]
We prove that if every bounded subset of $X^{*}$ is $w^{*}$-separable, $X$ is compactly locally uniformly convex, $X$ is 2-strictly convex and $X$ is nonsquare, then there exists a sequence $\{x_n\}_{n = 1}^\infty $ of dentable points of $B(X)$ such that $S(X) \subset \mathop \cup _{n = 1}^\infty B(x_n,{r_n})$, where $r_{n}< 1$ for all $n\in N$. Moreover, we also prove that if $A$ is a bounded closed convex subset of $X$, then $x\in A$ is a strongly exposed point of $A$ if and only if $x$ is a dentable point of $A$ and $x$ is a $w^{*}$-exposed point of $\overline {{A^{{w^*}}}}$.
Keywords: Compactly locally uniformly convex, ball-covering property, dentable point, nonsquare space, 2-strictly convex space.
MSC: 46B20
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