Locally \(2\)-uniform convexity and ball-covering property in Banach space (Q2341404)

The main results of the paper are: Theorem 2.1. Let \(X\) be a separable Banach space whose topological dual \(X^*\) is a locally 2-uniformly rotund space in the sense of \textit{F. Sullivan} [Can. J. Math. 31, 628--636 (1979; Zbl 0422.46011)]. Then for every \(\varepsilon \in (0, 1)\), there exist sequences \(\{x_{n,1}^*\}_{n \in \mathbb N}\) and \(\{x_{n,2}^*\}_{n \in \mathbb N}\) of strongly extreme points of \(B_{X^*}\) such that \[ S_{X^*} \subset \bigcup_{n \in \mathbb N}\left(B(x_{n,1}^*, 1 - \varepsilon/8) \cup B(x_{n,2}^*, 1 - \varepsilon/8) \cup B((x_{n,1}^* + x_{n,2}^*)/2, \varepsilon)\right). \] Theorem 2.5. Let \(X\) be a separable Banach space which is at the same time locally 2-uniformly rotund and uniformly non-square. Then there exist a sequence \(\{x_n\}_{n \in \mathbb N}\) of strongly extreme points of \(B_{X}\) and a sequence \(\{r_n\}_{n \in \mathbb N} \subset (0,1)\) such that \(S_{X} \subset \bigcup_{n \in \mathbb N}B(x_n, r_n)\).