Minimal Ball-Covering Of The Unit Spheres In N-dimensional Normed Spaces

A familyβof (open)closed balls in a Banach space X is said to be a ball-covering of X if every ball inβdoes not contain the origin in its interior and whose union covers the unit sphere S_X of X ; a ball-coveringβis said to be minimal if the cardinal ofβis less than or equal to cardinal of every ball-covering of X . Article [8] show that a norming set A of an n-dimensional space X has at least n + 1 elements ; that B_X^* has at least 2n exposed points which forms a symmetric norming set of X , and that B_X has exactly 2n exposed points if and only if X is isometric to l₁ⁿ. Based on these facts , this paper proves that (1) ifβ_min^# = 2n,then X is isometric to (Rⁿ,||·||_∞); (2)Ifβ_min^# ( X )= n + l, and {e_i}_i=1^∞(?) exp B_X^*,then (â…°) for each element x^*∈exp B_X, there are at least l－1 components of x^* that are 0 (â…±) for any two exposed points x₁^* ,x₂^* in B_X^*, there are at most n－l+ 2 non-zero components; and (â…²) if there exists x^*∈exp B_X^* such that its n－l+ 1 components are not 0, then there exists a l-dimensional subspace X_l (?) X that is isometric to ( R^l,||·||_∞).