Minimal Ball-Covering Of The Unit Spheres In R～n

A family β of closed balls in a Banach space X is a ball-covering of X if every ball in β does not contain the origin in its interior and whose union covers the unit sphere S_x of X, and a ball-covering β is said to be minimal if the cardinal of β is less than or equal to the cardinal of every ball-covering of X. Article[1] showed that a n-dimensional Banach space admits a minimal ball-covering not less than n+1 balls and further,if X is smooth ,in particular,X = Rⁿ,then the sphere can be covered by n+ 1 balls. Furthermore,article [2] gives that the radius of minimal ball-coverings is more than or equal to n/2 and it is attained whenever the centers of the n+1 balls of a minimal ball-covering are the vertices of a regular inscribed n-simplex of the sphere n/2S_x In such basis,this article first proves that there exists a specific ball-covering with the smallest radius in Rⁿ if a set {x_i}_i=1<sup>m satisfying some given term, then presents a minimal ball-covering with arbitrary given r ≥3^1/2/2 as its radius.