Ball-covering Properties Of Banach Spaces

By a ball-coveringβ= {B_r }(?) of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have ball-covering property, if it admits a ball-covering consisting of countable many balls. A ball-coveringβis said to be minimal if the cardinal ofβis less than or equal to the cardinal of every ball-covering of X. Article [7] showed that if dimX = n, then n + 1≤β_min^#≤2n ; if, in addition, X is smooth, thenβ_min^#=n+1；β_min^#=2n if and only if X is isometric to (Rⁿ,‖‖_∞);If X is a Gateaux differentiability space or a locally uniformly convex space, then the unit sphere admits such a countable ball-covering if and only if X^* is w^* -separable. By a lemma of [8], this paper presents that ifβ_min^#=2n-1 , then there is an n-1 dimensional subspace of X isometric to (R^n-1,‖‖_∞). Moreover, on this basis , this text combines article [11], first step's discussing the relation of the hereditarily indecomposable space and the ball-covering property space.