Locally $2$-uniform convexity and ball-covering property in Banach space

Abstract

In this paper, we prove that if $X$ is a separable space and $X^{*}$ is a locally $2$-uniform convex space, then for any $\varepsilon \in (0,1)$, there exist sequences $\{x_{n,1}^*\}_{n = 1}^\infty $ and $\{x_{n,2}^*\}_{n = 1}^\infty $ of strongly extreme points such that \[ \mathop \cup \limits_{n = 1}^\infty \left\{B(x_{n,1}^*,1 - \frac{1}{8}\varepsilon) \cup B(x_{n,2}^*,1 - \frac{1}{8}\varepsilon ) \cup B(\frac{{x_{n,1}^* + x_{n,2}^*}}{2},\varepsilon )\right\} \] is a ball-covering of $X^{*}$. Moreover, we also prove that if (1) $X$ is a separable space; (2) $X$ is a locally $2$-uniform convex space; (3) $X$ is a uniformly nonsquare space, then there exists a sequence $\{x_n\}_{n = 1}^\infty $ of strongly extreme points such that $\mathop \cup \limits_{n = 1}^\infty B(x_n,{r_n})$ is a ball-covering of $X$.