This paper presents two counterexamples about ball-coverings of Banach spaces and shows a new characterization of uniformly non-square Banach spaces via ball-coverings.
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 of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.
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.
CHENG LiXin, SHI HuiHua & ZHANG Wen School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
