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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

Yunan Cui
Dept. of Mathematics, Harbin University of Science and Technology, Harbin 150080, P. R. China


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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