Ball-covering Properties Of Banach Spaces

The whole Banach space geometry is a geometry about the unit ball andunit sphere of Banach spaces.Even among other knowledge branches,the directuses of"ball"to study other aspects of knowledge became important parts of thecorresponding branches.For instance,the Mazur sintersectionproperty which be-longs to Banach space geometry,Plank problem in complex analysis,topologyproblem in non-linear analysis,Packing problem in optimization theory andso on.Article[20] proposes a bran-new angle of view,to study on"how manyballs,which do not contain the origin in their interiors,can the unit sphere ofa Banach space be covered by". So does this paper starts from,to study thecardinal of ball-coverings of Banach spaces and the radius of a ball-covering.By the relationships between the n-simplex in an n-dimension space and itscircumscribed sphere(see,article[23]),this paper presents that for all n≥2 thesmallest radius of all minimal ball-coverings of Rn is n2, and it can be attainedwhenever the n+1 centers of the n+1 balls of a minimal ball-covering are thevertices of a regular inscribed n-simplex of the sphere n2SX.Then this paper makesa certain of another interesting problem that occurred during the former proof:forany fix constant r,there is a ball-covering whose covering radius is r and which isa minimal ball-covering among all the ball-coverings with radius r.