A Characterization Of Uniformly Non-square Banach Spaces Via Ball-coverings

By a ball-covering B of a Banach space X, We mean that it is a collection of open(closed) balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have ball-covering property, if it admits a ball-covering consisting of countably many balls; and a ball-covering B is called minimal, if its cardinal number B^# is smallest among all the cardinal numbers of ball-coverings of X. Banach spaces X is called uniformly non-square, if (R²,‖·‖_∞) can not be represented in X, that is, there existsε> 0 such that for every two dimension subspace X₂ of X, if T : X₂→(R²,‖·‖_∞) is a linear isomorophic, then‖T‖‖T^-1‖≥1 +ε. This paper presents a new characterization of uniformly non-square Banach space via ball-coverings of Banach spaces: Banach spaces X is uniformly non-square if and only if there existα,β>0 such that for every two dimensional subspace X_? of X, there is a ball-covering B of X(?) satisfying B^# = 3; r(B)≤βand B isα-off the origin.