A normed space is said to have ball-covering property if its unit sphere can be contained in the union of countably many open balls off the origin. This paper shows that for every ε > 0 every Banach space with a w * -separable dual has a 1+ε-equivalent norm with the ball covering property.Citation: Cheng L X, Shi H H, Zhang W. Every Banach space with a w * -separable dual has a 1 + ε-equivalent norm with the ball covering property.

Abstract. By a ball-covering B of a Banach space X, we mean a collection of open balls off the origin in X and whose union contains the unit sphere of X; a ball-covering B is called minimal if its cardinality B # is smallest among all ball-coverings of X. This article, through establishing a characterization for existence of a ball-covering in Banach spaces, shows that for every n ∈ N with k ≤ n there exists an n-dimensional space admitting a minimal ball-covering of n + k balls. As an application, we give a new characterization of superreflexive spaces in terms of ball-coverings. Finally, we show that every infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space possessing a countable ball-covering.1. Introduction. The study of geometric and topological properties of unit balls of normed spaces plays a central rule in the geometry of Banach spaces. Almost all properties of Banach spaces, such as convexity, smoothness, reflexivity, the Radon-Nikodým property, etc., can be viewed as properties of the unit ball. We should also mention here several topics concerning the behavior of families of balls, for example, the Mazur intersection property (see, for instance, [14] Starting from a different viewpoint, this article is devoted to studying the behavior of families B of open balls off the origin in a Banach space X whose union contains the unit sphere of X. We call such a family B a ballcovering of X. This notion was first introduced in [3]. For a ball-covering B ≡ {B(x i , r i )} i∈I of X, we denote by B # its cardinality and by r(B) the least upper bound of the radius set {r i } i∈I , and we call it the radius of B. We say that a ball-covering is minimal if its cardinality is the smallest of

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