A note on ball-covering property of Banach spaces

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 = ℓ 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 .