Zhang, Wen, Characterizations of universal finite representability and b-convexity of Banach spaces via ball coverings, Acta mathematica Sinica. English series (Print) , 28(7), 2012, pp. 1369-1374
By a ball-covering B of a Banach space X, wemean 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.