Ball-Covering Property Of Banach Space

Up to now,many researches on the geometric theory of Banach space are based on the unit sphere or unit sphere.Not only are many geometric properties of space inseparable from unit sphere,but also many researches take sphere as the research object,so this paper studies the property of Banach spaces.First,we prove that what kind of Banach space is uniform non-l4(1)space,namely:if there exists two constants α,β>0 such that for every four dimensional subspace Y of X,there is a ball-covering B of Y satisfying c(B)<c(R4,||·||),B is a-off the origin and r(B)≤β,Banach space X is uniformly non-l4(1)space.Then we prove that if B(Xi)is closed convex hull of strongly exposed points of B(Xi),and Xi is separable for any i∈N,then for any α∈(0,1),there exists a ball-covering B={B(xn,rn)}(?)of l∞(Xi*),such that B is a-off the origin.Finally we prove that if Xi is separable for any i∈N,there existα,β>0 and a ball-covering B＝{B(xn*,rn)}(?)of l∞(Xi)such that B is α-off the origin and r(B)≤β.