Dual Banach spaces with the ball-covering property

Abstract

We study ball-covering properties of dual Banach spaces and their connections with the geometry of predual unit balls. One of our main results shows that, for every separable Banach space $X$, the unit ball $B_X$ is a slicely countably determined set if and only if $\operatorname{bc}(X^*)=1$, where $\operatorname{bc}(\cdot)$ is the ball-covering index introduced by A. J. Guirao, A. Lissitsin, and V. Montesinos. We obtain several sufficient conditions for the uniform ball-covering property in dual spaces, including duals of spaces with a $K$-unconditional basis for $K<2$, and duals of separable spaces whose unit ball is the closed convex hull of a set of uniformly strongly exposed points. The constant $2$ is sharp: there is a space with a $2$-unconditional basis whose dual fails the ball-covering property. Applications are given to spaces of operators and to Lipschitz spaces. In particular, $\mathcal L(L_p[0,1])$ has the uniform ball-covering property for every $1<p<\infty$, which answers a question posed by Q. Bao, R. Liu, and J. Shen. As an application to Lipschitz spaces, we prove that $\operatorname{Lip}_0(M)$ has the uniform ball-covering property whenever $M$ is a separable complete ultrametric or H\"older metric space.