Ball-covering property of Banach spaces

Covering spheres of Banach spaces by balls

“…It easily follows from the separation theorem that, if S X admits a countable covering by balls (even by closed convex sets) which do not contain the origin, then the dual space X * contains a countable total set, i.e. X * is w * -separable [1]. Moreover, in [1] the Author claims that the converse is true under some additional assumptions on the space X , like Gateaux smoothness or local uniform rotundity (see also Zbl 1139.46016).…”

“…X * is w * -separable [1]. Moreover, in [1] the Author claims that the converse is true under some additional assumptions on the space X , like Gateaux smoothness or local uniform rotundity (see also Zbl 1139.46016). The constructions provided there give coverings by balls whose radii are not uniformly bounded.…”

Characterizations of universal finite representability and b-convexity of Banach spaces via ball coverings

“…This notion was introduced by Cheng [1] and intensively studied in [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. It was shown in [1] that every Banach space X with ball-covering has a w * -separable dual. Conversely there exists a Banach space with a w * -separable dual which does not admit ball-covering property in general.…”

Minimal ball-coverings in Banach spaces and their application

“…This can be directly obtained from Theorem 3.1 of [3], where we prove that every Banach space admitting a countable ball-covering with radii at most r < 1 is separable.…”

“…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.…”