Problems On Minimal Ball-covering Of L₁ⁿ

We call B?{Bi}i?I is a ball-covering of Banach space X,if each ball Bi is off the origin and the union of {Bi}i?Icontains the unit sphere SX.We say that B is a minimal ball-covering,if the cardinality of the ball-covering is less than or equal to the cardinality of the whole ball-coverings.In this paper,we consider two problems about the ball-covering in the space l1n:one is the cardinality of the minimal ball-covering of l1n,and the other is the optimal radius of the minimal ball-covering of l1n.For the first problem,this paper improves on the existing results of n+2 by Hu Zhifang and Zhao Xin,and finally obtains the following results:When n=2,the cardinality of the minimal ball-covering k=4;When n.>2,it is n+1.For the second problem,this article has not fully solved it,but shows all the thinking processes and ideas in detail,and gets an upper bound of l1n for the optimal radius of the minimal ball-covering of l1n.