A note on ball-covering property of Banach spaces

Abstract

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere S X of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X * of X is w * separable, then for every ε > 0 there exist a (1 + ε)-equivalent norm on X, and an R > 0 such that in this new norm S X admits a ball-covering by countably many balls of radius R.Namely, we show that R = R(ε) can be taken arbitrarily close to (1 + ε)/ε, and that for X = 1 [0, 1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε → 0.

Some geometric and topological properties of Banach spaces via ball coverings

“…Therefore, this combining with results of [6,[8][9][10] implies that a Banach space X can be renormed such that every point is a b X -G δ point if and only if X * is w * -separable; and that if X admits the ball-covering property, then every b X -compact set is sequentially compact.…”

“…We say that X has the ball-covering property if it admits a ball-covering of countably many balls. In the recent years, the study of the ball-covering property of Banach spaces has also brought mathematicians attention, and such property has been intensively studied in [3][4][5][6][7][8]11,[13][14][15][16]20,21].…”

A characterization of Banach spaces containing $\ell_1(κ)$ via ball-covering properties

“…It is known that separable Banach spaces have the BCP, but the converse is not true, because ℓ ∞ has the BCP. In recent years the BCP of Banach spaces has been studied by various authors (see e.g., [7], [8], [10], [15] and [22]).…”

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

“…We say that X has the ball-covering property if it admits a ball-covering of countably many balls. 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.…”